Error Compensation of Workpiece Localization
Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
Error Compensation of Workpiece Localization Z.H. Xiong and Z.X. Li Dept.of EEE, Hong Kong University of Science & Technology, China (e-mail: [email protected], fax:(852)2358-1485)
Abstract
by many researchers to solve the minimization problem of the objective function [1] [2] [3] [4]. In order to get the points set fy g, high precision measurement should be taken on the workpiece surfaces. Two kind of techniques, namely, contact probing and non-contact sensing, can be applied to get dimensional measurement data from a 3D workpiece. Non-contact sensing utilize laser, optics and CCD sensors and can obtain a large amount of data within a short time. These methods are sensitive to shiny and dark surfaces and usually have poor accuracy [6]. Laser digitizers also su er from diculties of calibration and limited accessibility to concavities or undercuts that extend beyond the probing angles. Compared with non-contact sensing, contact probing provides higher accuracy and resolution, and has widely available operator skills [7]. So, contact probing usually is the rst-choice solution for metrology practitioners. The touch trigger probe, which is also called a switching probe or touch probe, provides an economical method of on-machine metrology on tool machines as well as on coordinate measuring machines (CMM). The touch trigger probe has become one of the basic building blocks for supporting untended machining in manufacturing systems. In spite of the high precision of the touch trigger probe, there are errors associated with touch probing applications. Two sources of errors, speci cally, pretravel or lobing variations [8] [9] [10] and errors due to probe radius [11] [12] [6] [7] [13] [14] [15] [16], are reported. When the touch trigger probe is in contact with the workpiece, the contact force between them will build up gradually. A trigger signal will be generated when the force exceeds a threshold in the sensing system. The signal is used by CMM or tool machine to record the point coordinates at the triggered instant. However, after the touch instant the probe continues to travel in the probe approaching direction until the trigger signal is generated. The travel distance between the touch instant and the trigger instant is know as probe pretravel. It is known that pretravel error is mainly caused by bending de ection of the stylus shaft
Workpiece localization has direct relations with many manufacturing automation applications. In order to gain accurate workpiece measurement by coordinate measuring machines (CMM) or on-machine measurement system, the touch trigger probe is widely adopted. In spite of the high repeatability of the touch trigger probe, there are still error sources associated with the probe. In this paper, we will focus on probe radius compensation. Several compensation methods in related papers are reviewed. In addition, a new radius compensation method is proposed in this paper. Simulation and experimental results of probe radius compensation by di erent methods are given. It is shown that our proposed method has the best performance both in terms of compensation accuracy and computational time. The method is also implemented in a computer aided setup (CAS) system.
i
1 Introduction With direct applications in manufacturing automation like workpiece setup, dimensional inspection, robot assembly, etc., workpiece localization has received much attention from many researchers [1] [2] [3] [4]. Given a set of measurement points y 2 R3 , i = 1; ; n, with corresponding home surface S , the problem is formulated as a least squares problem with the objective function given by i
i
"(g; x1 ; ; x ) = n
X k y ; gx k2 n
=1
i
i
(1)
i
where g 2 SE (3) is the Euclidean transformation transforming the CAD frame of the workpiece to some known machine reference frame, and x 2 R3 is the home point of y 2 R3 on the corresponding surface S [5]. Many iterative algorithms have been proposed i
i
i
This research is supported in part by RGC Grant No. HKUST 6220/98E and CRC 98/01.EG02
1 0-7803-6475-9/01/$10.00 © 2001 IEEE
2249
2 Probe Radius Compensation
and is greatly a ected by the measuring speed [10]. In the cited papers, the authors have given di erent models to predict the pretravel error. When the probe stylus touches the workpiece sur-
Many papers relative to probe radius compensation are from reverse engineering problem [6] [13] [15] [14], which refers to reconstruct a part model from a physical engineering part or a surface from measurement points on a real part/surface. In [15], the point set fy0 g is rst used to generate a probe center surface S , then nd the corresponding normal vector N at the point y0 . After that, the o set point P on the compensated surface is estimated by i
Stylus
p
r
y’i
p;i
ni yi
i
c;i
Si
P = P + RN c;i
CW
y’i ni
(2)
p;i
where, R is the radius of the probe stylus. The compensated surface S will be generated using the P points. In [6], an iterative method has been applied to re ne the normal directions. Then, new compensated points and surfaces can be generated. In reverse engineering applications, CAD model is not available. The normal directions are just approximations of real normal directions at the contact points. Therefore, the compensated points may not lie on the surfaces. In [13], proper and improper probe directions have
Stylus
11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 yi
p;i
Si
c
Figure 1: Probe error caused by stylus radius
face, the coordinates recorded by the system are the coordinates of the center point y0 of the probe tip. As shown in gure 1, the exact contact point between the stylus and the workpiece is point y . If we directly use the coordinates of point y0 as the coordinates of the contact point y , signi cant errors will be introduced, since the radius of the probe is usually several millimeters. This error is known as probe radius error. Nowadays, the repeatability of the touch trigger probe is rather high. The probe used by us is a MP12 probe from Renishaw. Its speci cations show that repeatability of 10m is valid for test velocity of 480mm=min at stylus tip with a 50mm long stylus and trigger force of 65 ; 85g. It is also known that the pretravel error of the touch trigger probe is usually several ten microns [8] [10]. We will only address the error caused by probe radius in this paper, with the assumption that the pretravel error can be omitted in our case. The remainder of the paper is organized as follows. In section 2, we will review the radius compensation methods in related papers. We will also propose a compensation method for the localization algorithms. In section 3, we will do some simulations on a cube and a model with sculptured surface. Three di erent methods will be used. Comparison of compensation results will show the eciency of our proposed compensation method. In section 4, a real workpiece will be used to verify our proposed method. Finally, we will reach a conclusion in section 5. i
c;j
improper direction
proper direction CW
i
y’i Stylus
y’i
Si
yi
11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111
i
yi n i
i
ni
Figure 2: Classi cation of surface probing directions been pointed out. When the probing direction coincides with the normal vector of the contact point, we can compensate for the probe radius error using the probing direction as the normal vector in equation (2). Thus, the error will be minimized. Otherwise, we call it an improper direction. Since the normal vectors of the contact points can not be determined beforehand, the authors applied a procedure based on the iterative measurement and re nement of probing direction [13]. Therefore, the processing time to obtain the accurate points set is relatively longer. In [14], a three-point measuring scheme has been adopted. The aim of this method is to estimate the surface normal vector by probing three points, which are not co-linear, around the sampling point. Since these three points form a facet around the sampling point, we can approximate the normal vector of the 2
2250
fy g is used to get the optimal transformation g +1 . Then, a new set of fy g will be compensated for using
sampling point with the normal vector of the facet. When the workpiece surface is relatively at, this method is e ective. However, for sculptured surfaces, the approximated normal vectors may deviate from the actual normal vectors. For workpiece localization, a direct compensation method has been proposed in [12] and [11]. The authors have given formulas to compensate for the probed points that were sampled from di erent directions. The components of the normal vectors are still needed in their method. We need to estimate the normal vectors either by the three point measuring scheme in [14], or by a specially designed sensor called a spherical potentiometer in [17]. By using the sensor, the contact point on the probe stylus can be determined. Thus, the compensation for the probe radius is easily done. A disadvantage of this method is that the workpiece need to be conductor, since the workpiece will be a negative electrode of the source. New hardware and calibration work are also needed. Instead of directly compensating for the probed points, there are methods that use the probe center points set fy0 g [16] [7]. It is noted that all the points y0 s lie on o set surfaces of the workpiece. In [7], the aim of registration is just like that of workpiece localization. Since CAD model is known in this case, the author rst get o set surfaces from the CAD model. Then, the ICP (iterative closest point) method will be implemented to solve registration problem. NURBS (non-uniform rational B-spline) is chosen to represent the CAD model surfaces. It has been proven that the o set surface of a NURBS surface is generally not a NURBS [18]. The author used linear least squares tting to get approximated NURBS surfaces. To avoid the divergence of the ICP method, closer di erence between CAD frame and the workpiece is needed. In the simulation results they gave, translational and rotational di erence are within the range of 1mm and 1 respectively. Unlike the ICP method in [7], some ecient localization algorithms have been adopted in [16]. These algorithms have much bigger convergence ranges [19]. Probe radius compensation method has also been proposed in [16]. The compensation procedure is as follows. First, we get the optimal transformation g using the probe center points set fy0 g and nominal surfaces S s. Then, compensate for y0 by
g +1 as g in equation (2). This procedure will continue until a desired precision is reached between g and g +1 . i
k
k
In order to accurately compensate for the probe ra-
g
Stylus
Stylus
yi
Figure 3: Point relations in probe radius compensation dius error based on the workpiece model, we propose a new method in this paper. Since the probed points y0 s are on the o set surfaces of the workpiece, we can modify the objective function in equation (1) by i
"(g; x1 ; ; x ) = n
i
i
=1
i
(4)
i
i
i
i
i
i
i
i
x0 = x + rn i
i
(5)
i
Then, the objective function will have the same form as in equation (1).
"(g; x01 ; ; x0 ) = n
X k y0 ; gx0 k2 n
=1
i
i
(6)
i
Now, we can utilize the Tangent or Hong-Tan algorithms in [1] [16] to get the optimal transformation. Compared with the method in [7], we do not need to get the explicit o set surface representations. So, the computational workload is smaller and the surface errors caused by linear least squares tting are eliminated. The localization algorithms we adopted also have better convergence ranges. As for the compensation method in [16], we always need apply Tangent or Hong-Tan algorithm for several times. Our proposed
where n is the normal vector at the home point x of point y0 , r is the radius of the probe, y is the compensated point of y0 . After one compensation, the set i
i
i
n
where x 2 R3 is the home surface point of the contact point y , as shown in gure 3. n 2 R3 is the unit outward normal vector at x on nominal surfaces. n0 = gn is the normal vector in machine frame. If we de ne the probe center point x0 in the CAD frame corresponding to y0 as:
(3)
i
X k y0 ; g(x + rn ) k2 i
i
i
Si
CW
i
k
r
y’i n ’i
k
i
Si
ni xi
o
i
r x ’i
i
y = y0 ; rg n
k
k
i
i
k
i
i
3 2251
method just requires to apply the algorithm one time. So it has a higher computational eciency. Moreover, as shown in gure 3, our method has clear geometric correspondence between the probe center points y0 s and the home surface points x s. We will show the simulation and experimental results in the remaining part of this paper. i
i
(a)
3 Simulation Results
(b)
Figure 4: Localization results with and without probe radius compensation
In this section, we will verify our proposed method with some simulation results. In the remainder of this paper, we will call the compensation method in [16] as method a, the compensation method we proposed in last section as method b. A PII 400 PC is used in our simulation. A cube will rst be used in the simulation. We generate measurement points on three orthogonal at surfaces by sampling 100 points on each surface and adding probe radius o set in the normal directions. The probe radius used in this simulation is 3mm. Then we transform the generated points randomly 30 times as in [7]. The applied transformations include translation and rotation parts. Unlike in [7], the translation and rotation values we choose are uniformly distributed random numbers within the range of 100mm and 30 respectively. To simulate the sampling errors in real application, we introduce some normally distributed random noise to the transformed points. For each generated points set, we apply three compensation methods, namely, no compensation method, method a and method b. We will compare the localization error and computational time for these three methods. Let g 2 SE (3) be the applied transformation, g 2 SE (3) be the optimal result given by localization algorithms. Then, g;1g = fR; pg 2 SE (3) is the transformation di erence between them. From p, we can get the translation errors along three axes. It is well known that the exponential map is surjective onto SO(3) [20]. So, for the rotation matrix R, we can nd ! 2 R3 , k ! k= 1 and 2 R such that R = exp(^!). Then we can use as representation of the rotation error in our simulation. In table 1, we show the statistical compensation results of these three methods. We can nd that the method without probe radius compensation has the biggest translation errors. This is easy to see from gure 4, since all points are localized onto the nominal cube surfaces as in (b), instead of being localized onto the o set surfaces as in (a). From the table, we can see that method b gives the best results among the three
methods. Method a always takes more computational time, since it needs more iterations than the other two methods. A model with sculptured surface, as shown in gure 5,
o
Figure 5: A model with a sculptured surface in our simulation has also been used in our simulation. We just use the sculptured surface and a vertical surface at this time. Compensation results are shown in table 2. We nd that method b has the fastest convergence rate now and that its compensation results has also the best performance.
4 Experimental Results In this section, we will get our measurement data from a real workpiece, as shown in gure 6. Auto probing routine has been applied to measure the surfaces as in the experiment. The system measures 36 points for each selected surface. The actual probe radius in our system is 2:998mm. We will compare the localization and compensation results of the three methods. Figure 7(a) shows the probed points of the four surfaces. Figure 7(b) shows the localization results after our radius compensation. Since we do not know the actual transformation, we can not get the transformation errors as in the last section. We will compare 4
2252
Table 1: Simulation results of probe radius compensation of a cube Mean value Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) No compensation 2.578e-4 2.9996 3.0040 3.0030 0.1587 Method a 2.578e-4 1.1271 0.5421 0.6058 0.6580 Method b 2.578e-4 -0.0004 0.0040 0.0030 0.1627 Standard deviation (deg) Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) No compensation 7.782e-5 0.0004 0.0010 0.0011 0.0224 Method a 7.782e-5 0.6027 0.4668 0.4769 0.0873 Method b 7.782e-5 0.0004 0.0010 0.0011 0.0151
(deg)
Table 2: Simulation results of probe radius compensation of a sculptured model Mean value
(deg) Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s)
No compensation 0.5950 Method a 0.0123 Method b 0.0043
(deg) No compensation 0.0040 Method a 0.0064 Method b 0.0047
1.1466 -0.0301 -0.0025
-2.5891 -1.1661 0.2570 -0.0147 -0.0023 0.6890 0.0007 -0.0045 0.1457 Standard deviation Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) 0.0006 0.0005 0.0010 0.0377 0.0192 0.0080 0.0210 0.2302 0.0007 0.0005 0.0043 0.0260
Figure 6: Experimental model
(a)
the computational time and the residue errors of the localization algorithms, which can evaluate how the probed data ts the model. From table 3, we can see that the residue error of method b is the smallest and its computational time is also the shortest.
(b)
Figure 7: Experimental model and localization results methods have been reviewed. We have proposed a new compensation method for workpiece localization application. We have compared the performance of our proposed method with other methods. It has been shown that our proposed method has the best performance. The compensation method has been applied in our computer aided setup system.
5 Conclusion In this paper, we have addressed the probe radius compensation problem. Di erent kinds of compensation 5
2253
[13] C. MENG and F.L. CHEN. Curve and surface approximation from CMM measurement data. Computers ind. Engng, 30(2):211{25, 1996. [14] S.H. SUH, S.K. LEE, and J.J. LEE. Compensating probe radius in free surface modeling with CMM: Simulation and experiment. Int. J. Prod. Res., 34(2):507{23, 1996. [15] J R R Mayer, Y A Mir, and et al. Touch probe radius compensation for coordinate measurement using kriging interpolation. Proc. Instn. Mech. Engrs., 211(Part B):11{18, 1997. [16] Yunxian Chu. Workpiece Localization: Theory, Algorithms and Implementation. PhD thesis, HKUST, March 1999. [17] H. Aoyama, M. Kawai, and T. Kishinami. A new method for detecting the contact point between a touch probe and a surface. Annals of the CIRP, 38(1):517{20, 1989. [18] W. Tiller L. Piegl. The NURBS Book. Springer-Verlag, 1st edition, 1995. [19] Y.X. Chu, J.B. Gou, and Z.X. Li. Localization algorithms: Performance evaluation and reliability analysis. In Int. Conf. on Robotics and Automation, volume 4, pages 3652{ 7, 1998. [20] R. Murray, Z.X. Li, and S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.
Table 3: Experimental results of probe radius compensation No compen. Method a Method b
Res. error(mm) Comp. Time(s) 84.2410 0.71 32.1873 2.32 11.3610 0.66
Acknowledgment The authors would like to thank Mr. G.F. Liu and Miss H. Cheng for their helpful discussion.
References
[1] Z.X. Li, J.B. Gou, and Y.X. Chu. Geometric algorithms for workpiece localization. IEEE Transactions on Robotics and Automation, 14(6):864{78, Dec. 1998. [2] A.B. Forbes. Least-squares best- t geometric elements. Technical report, DITC 140/89, National Physical Laboratory, UK, 1989. [3] J. Hong and X. Tan. Method and apparatus for determining position and orientation of mechanical objects. U.S. Patent No.5208763, 1990. [4] H.T. Yau and C.H. Menq. A uni ed least-squares approach to the evaluation of geometric errors using discrete measurement data. Int. J. Mach. Tools Manufact., 36(11):1269{1290, 1996. [5] J.B. Gou, Y.X. Chu, and Z.X. Li. On the symmetric localization problem. IEEE Trans. on Robotics and Automation, 14(4):533{540, 1998. [6] Y.M. Chiang and F.L. Chen. Sculptured surface reconstruction from CMM measurement data by a software iterative approach. Int. J. Prod. Res., 37(8):1679{95, 1999. [7] I. Ainsworth, M. Ristic, and D. Brujic. Cad-based measurement path planning for free-form shapes using contact probes. Int. J. Adv. Manuf. Technol., 16:23{31, 2000. [8] Q. Yang, C. Butler, and P. Baird. Error compensation of touch trigger probes on CMM. Measurement, 18(1):47{57, 1996. [9] W. Tyler Estler, S.D. Phillips, and et al. Error compensation for CMM touch trigger probes. Precision Engineering, 19(2/3):85{97, 1996. [10] Yin-Lin Shen and Xiaoping Zhang. Pretravel compensation for horizontally oriented touch trigger probes with straight styli. Journal of Manufacturing Systems, 18(3):175{86, 1999. [11] B. Kang, J.B. Gou, Y.X. Chu, and Z.X. Li. A cad-based probing and localisation method for arbitrarily xed workpiece. In Int. Conf. on Robotics and Automation, volume 2, pages 1259{64, 1997. [12] W L R Ip and M Loftus. Adaptive and compensation method in free-form surface assessment. Proc. Instn. Mech. Eng. Part B: Journal of Engineering Manufacture, 210:135{145, 1996.
6 2254
Error Compensation of Workpiece Localization Z.H. Xiong and Z.X. Li Dept.of EEE, Hong Kong University of Science & Technology, China (e-mail: [email protected], fax:(852)2358-1485)
Abstract
by many researchers to solve the minimization problem of the objective function [1] [2] [3] [4]. In order to get the points set fy g, high precision measurement should be taken on the workpiece surfaces. Two kind of techniques, namely, contact probing and non-contact sensing, can be applied to get dimensional measurement data from a 3D workpiece. Non-contact sensing utilize laser, optics and CCD sensors and can obtain a large amount of data within a short time. These methods are sensitive to shiny and dark surfaces and usually have poor accuracy [6]. Laser digitizers also su er from diculties of calibration and limited accessibility to concavities or undercuts that extend beyond the probing angles. Compared with non-contact sensing, contact probing provides higher accuracy and resolution, and has widely available operator skills [7]. So, contact probing usually is the rst-choice solution for metrology practitioners. The touch trigger probe, which is also called a switching probe or touch probe, provides an economical method of on-machine metrology on tool machines as well as on coordinate measuring machines (CMM). The touch trigger probe has become one of the basic building blocks for supporting untended machining in manufacturing systems. In spite of the high precision of the touch trigger probe, there are errors associated with touch probing applications. Two sources of errors, speci cally, pretravel or lobing variations [8] [9] [10] and errors due to probe radius [11] [12] [6] [7] [13] [14] [15] [16], are reported. When the touch trigger probe is in contact with the workpiece, the contact force between them will build up gradually. A trigger signal will be generated when the force exceeds a threshold in the sensing system. The signal is used by CMM or tool machine to record the point coordinates at the triggered instant. However, after the touch instant the probe continues to travel in the probe approaching direction until the trigger signal is generated. The travel distance between the touch instant and the trigger instant is know as probe pretravel. It is known that pretravel error is mainly caused by bending de ection of the stylus shaft
Workpiece localization has direct relations with many manufacturing automation applications. In order to gain accurate workpiece measurement by coordinate measuring machines (CMM) or on-machine measurement system, the touch trigger probe is widely adopted. In spite of the high repeatability of the touch trigger probe, there are still error sources associated with the probe. In this paper, we will focus on probe radius compensation. Several compensation methods in related papers are reviewed. In addition, a new radius compensation method is proposed in this paper. Simulation and experimental results of probe radius compensation by di erent methods are given. It is shown that our proposed method has the best performance both in terms of compensation accuracy and computational time. The method is also implemented in a computer aided setup (CAS) system.
i
1 Introduction With direct applications in manufacturing automation like workpiece setup, dimensional inspection, robot assembly, etc., workpiece localization has received much attention from many researchers [1] [2] [3] [4]. Given a set of measurement points y 2 R3 , i = 1; ; n, with corresponding home surface S , the problem is formulated as a least squares problem with the objective function given by i
i
"(g; x1 ; ; x ) = n
X k y ; gx k2 n
=1
i
i
(1)
i
where g 2 SE (3) is the Euclidean transformation transforming the CAD frame of the workpiece to some known machine reference frame, and x 2 R3 is the home point of y 2 R3 on the corresponding surface S [5]. Many iterative algorithms have been proposed i
i
i
This research is supported in part by RGC Grant No. HKUST 6220/98E and CRC 98/01.EG02
1 0-7803-6475-9/01/$10.00 © 2001 IEEE
2249
2 Probe Radius Compensation
and is greatly a ected by the measuring speed [10]. In the cited papers, the authors have given di erent models to predict the pretravel error. When the probe stylus touches the workpiece sur-
Many papers relative to probe radius compensation are from reverse engineering problem [6] [13] [15] [14], which refers to reconstruct a part model from a physical engineering part or a surface from measurement points on a real part/surface. In [15], the point set fy0 g is rst used to generate a probe center surface S , then nd the corresponding normal vector N at the point y0 . After that, the o set point P on the compensated surface is estimated by i
Stylus
p
r
y’i
p;i
ni yi
i
c;i
Si
P = P + RN c;i
CW
y’i ni
(2)
p;i
where, R is the radius of the probe stylus. The compensated surface S will be generated using the P points. In [6], an iterative method has been applied to re ne the normal directions. Then, new compensated points and surfaces can be generated. In reverse engineering applications, CAD model is not available. The normal directions are just approximations of real normal directions at the contact points. Therefore, the compensated points may not lie on the surfaces. In [13], proper and improper probe directions have
Stylus
11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 yi
p;i
Si
c
Figure 1: Probe error caused by stylus radius
face, the coordinates recorded by the system are the coordinates of the center point y0 of the probe tip. As shown in gure 1, the exact contact point between the stylus and the workpiece is point y . If we directly use the coordinates of point y0 as the coordinates of the contact point y , signi cant errors will be introduced, since the radius of the probe is usually several millimeters. This error is known as probe radius error. Nowadays, the repeatability of the touch trigger probe is rather high. The probe used by us is a MP12 probe from Renishaw. Its speci cations show that repeatability of 10m is valid for test velocity of 480mm=min at stylus tip with a 50mm long stylus and trigger force of 65 ; 85g. It is also known that the pretravel error of the touch trigger probe is usually several ten microns [8] [10]. We will only address the error caused by probe radius in this paper, with the assumption that the pretravel error can be omitted in our case. The remainder of the paper is organized as follows. In section 2, we will review the radius compensation methods in related papers. We will also propose a compensation method for the localization algorithms. In section 3, we will do some simulations on a cube and a model with sculptured surface. Three di erent methods will be used. Comparison of compensation results will show the eciency of our proposed compensation method. In section 4, a real workpiece will be used to verify our proposed method. Finally, we will reach a conclusion in section 5. i
c;j
improper direction
proper direction CW
i
y’i Stylus
y’i
Si
yi
11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111
i
yi n i
i
ni
Figure 2: Classi cation of surface probing directions been pointed out. When the probing direction coincides with the normal vector of the contact point, we can compensate for the probe radius error using the probing direction as the normal vector in equation (2). Thus, the error will be minimized. Otherwise, we call it an improper direction. Since the normal vectors of the contact points can not be determined beforehand, the authors applied a procedure based on the iterative measurement and re nement of probing direction [13]. Therefore, the processing time to obtain the accurate points set is relatively longer. In [14], a three-point measuring scheme has been adopted. The aim of this method is to estimate the surface normal vector by probing three points, which are not co-linear, around the sampling point. Since these three points form a facet around the sampling point, we can approximate the normal vector of the 2
2250
fy g is used to get the optimal transformation g +1 . Then, a new set of fy g will be compensated for using
sampling point with the normal vector of the facet. When the workpiece surface is relatively at, this method is e ective. However, for sculptured surfaces, the approximated normal vectors may deviate from the actual normal vectors. For workpiece localization, a direct compensation method has been proposed in [12] and [11]. The authors have given formulas to compensate for the probed points that were sampled from di erent directions. The components of the normal vectors are still needed in their method. We need to estimate the normal vectors either by the three point measuring scheme in [14], or by a specially designed sensor called a spherical potentiometer in [17]. By using the sensor, the contact point on the probe stylus can be determined. Thus, the compensation for the probe radius is easily done. A disadvantage of this method is that the workpiece need to be conductor, since the workpiece will be a negative electrode of the source. New hardware and calibration work are also needed. Instead of directly compensating for the probed points, there are methods that use the probe center points set fy0 g [16] [7]. It is noted that all the points y0 s lie on o set surfaces of the workpiece. In [7], the aim of registration is just like that of workpiece localization. Since CAD model is known in this case, the author rst get o set surfaces from the CAD model. Then, the ICP (iterative closest point) method will be implemented to solve registration problem. NURBS (non-uniform rational B-spline) is chosen to represent the CAD model surfaces. It has been proven that the o set surface of a NURBS surface is generally not a NURBS [18]. The author used linear least squares tting to get approximated NURBS surfaces. To avoid the divergence of the ICP method, closer di erence between CAD frame and the workpiece is needed. In the simulation results they gave, translational and rotational di erence are within the range of 1mm and 1 respectively. Unlike the ICP method in [7], some ecient localization algorithms have been adopted in [16]. These algorithms have much bigger convergence ranges [19]. Probe radius compensation method has also been proposed in [16]. The compensation procedure is as follows. First, we get the optimal transformation g using the probe center points set fy0 g and nominal surfaces S s. Then, compensate for y0 by
g +1 as g in equation (2). This procedure will continue until a desired precision is reached between g and g +1 . i
k
k
In order to accurately compensate for the probe ra-
g
Stylus
Stylus
yi
Figure 3: Point relations in probe radius compensation dius error based on the workpiece model, we propose a new method in this paper. Since the probed points y0 s are on the o set surfaces of the workpiece, we can modify the objective function in equation (1) by i
"(g; x1 ; ; x ) = n
i
i
=1
i
(4)
i
i
i
i
i
i
i
i
x0 = x + rn i
i
(5)
i
Then, the objective function will have the same form as in equation (1).
"(g; x01 ; ; x0 ) = n
X k y0 ; gx0 k2 n
=1
i
i
(6)
i
Now, we can utilize the Tangent or Hong-Tan algorithms in [1] [16] to get the optimal transformation. Compared with the method in [7], we do not need to get the explicit o set surface representations. So, the computational workload is smaller and the surface errors caused by linear least squares tting are eliminated. The localization algorithms we adopted also have better convergence ranges. As for the compensation method in [16], we always need apply Tangent or Hong-Tan algorithm for several times. Our proposed
where n is the normal vector at the home point x of point y0 , r is the radius of the probe, y is the compensated point of y0 . After one compensation, the set i
i
i
n
where x 2 R3 is the home surface point of the contact point y , as shown in gure 3. n 2 R3 is the unit outward normal vector at x on nominal surfaces. n0 = gn is the normal vector in machine frame. If we de ne the probe center point x0 in the CAD frame corresponding to y0 as:
(3)
i
X k y0 ; g(x + rn ) k2 i
i
i
Si
CW
i
k
r
y’i n ’i
k
i
Si
ni xi
o
i
r x ’i
i
y = y0 ; rg n
k
k
i
i
k
i
i
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method just requires to apply the algorithm one time. So it has a higher computational eciency. Moreover, as shown in gure 3, our method has clear geometric correspondence between the probe center points y0 s and the home surface points x s. We will show the simulation and experimental results in the remaining part of this paper. i
i
(a)
3 Simulation Results
(b)
Figure 4: Localization results with and without probe radius compensation
In this section, we will verify our proposed method with some simulation results. In the remainder of this paper, we will call the compensation method in [16] as method a, the compensation method we proposed in last section as method b. A PII 400 PC is used in our simulation. A cube will rst be used in the simulation. We generate measurement points on three orthogonal at surfaces by sampling 100 points on each surface and adding probe radius o set in the normal directions. The probe radius used in this simulation is 3mm. Then we transform the generated points randomly 30 times as in [7]. The applied transformations include translation and rotation parts. Unlike in [7], the translation and rotation values we choose are uniformly distributed random numbers within the range of 100mm and 30 respectively. To simulate the sampling errors in real application, we introduce some normally distributed random noise to the transformed points. For each generated points set, we apply three compensation methods, namely, no compensation method, method a and method b. We will compare the localization error and computational time for these three methods. Let g 2 SE (3) be the applied transformation, g 2 SE (3) be the optimal result given by localization algorithms. Then, g;1g = fR; pg 2 SE (3) is the transformation di erence between them. From p, we can get the translation errors along three axes. It is well known that the exponential map is surjective onto SO(3) [20]. So, for the rotation matrix R, we can nd ! 2 R3 , k ! k= 1 and 2 R such that R = exp(^!). Then we can use as representation of the rotation error in our simulation. In table 1, we show the statistical compensation results of these three methods. We can nd that the method without probe radius compensation has the biggest translation errors. This is easy to see from gure 4, since all points are localized onto the nominal cube surfaces as in (b), instead of being localized onto the o set surfaces as in (a). From the table, we can see that method b gives the best results among the three
methods. Method a always takes more computational time, since it needs more iterations than the other two methods. A model with sculptured surface, as shown in gure 5,
o
Figure 5: A model with a sculptured surface in our simulation has also been used in our simulation. We just use the sculptured surface and a vertical surface at this time. Compensation results are shown in table 2. We nd that method b has the fastest convergence rate now and that its compensation results has also the best performance.
4 Experimental Results In this section, we will get our measurement data from a real workpiece, as shown in gure 6. Auto probing routine has been applied to measure the surfaces as in the experiment. The system measures 36 points for each selected surface. The actual probe radius in our system is 2:998mm. We will compare the localization and compensation results of the three methods. Figure 7(a) shows the probed points of the four surfaces. Figure 7(b) shows the localization results after our radius compensation. Since we do not know the actual transformation, we can not get the transformation errors as in the last section. We will compare 4
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Table 1: Simulation results of probe radius compensation of a cube Mean value Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) No compensation 2.578e-4 2.9996 3.0040 3.0030 0.1587 Method a 2.578e-4 1.1271 0.5421 0.6058 0.6580 Method b 2.578e-4 -0.0004 0.0040 0.0030 0.1627 Standard deviation (deg) Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) No compensation 7.782e-5 0.0004 0.0010 0.0011 0.0224 Method a 7.782e-5 0.6027 0.4668 0.4769 0.0873 Method b 7.782e-5 0.0004 0.0010 0.0011 0.0151
(deg)
Table 2: Simulation results of probe radius compensation of a sculptured model Mean value
(deg) Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s)
No compensation 0.5950 Method a 0.0123 Method b 0.0043
(deg) No compensation 0.0040 Method a 0.0064 Method b 0.0047
1.1466 -0.0301 -0.0025
-2.5891 -1.1661 0.2570 -0.0147 -0.0023 0.6890 0.0007 -0.0045 0.1457 Standard deviation Trans X(mm) Trans Y(mm) Trans Z(mm) Comp. Time(s) 0.0006 0.0005 0.0010 0.0377 0.0192 0.0080 0.0210 0.2302 0.0007 0.0005 0.0043 0.0260
Figure 6: Experimental model
(a)
the computational time and the residue errors of the localization algorithms, which can evaluate how the probed data ts the model. From table 3, we can see that the residue error of method b is the smallest and its computational time is also the shortest.
(b)
Figure 7: Experimental model and localization results methods have been reviewed. We have proposed a new compensation method for workpiece localization application. We have compared the performance of our proposed method with other methods. It has been shown that our proposed method has the best performance. The compensation method has been applied in our computer aided setup system.
5 Conclusion In this paper, we have addressed the probe radius compensation problem. Di erent kinds of compensation 5
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[13] C. MENG and F.L. CHEN. Curve and surface approximation from CMM measurement data. Computers ind. Engng, 30(2):211{25, 1996. [14] S.H. SUH, S.K. LEE, and J.J. LEE. Compensating probe radius in free surface modeling with CMM: Simulation and experiment. Int. J. Prod. Res., 34(2):507{23, 1996. [15] J R R Mayer, Y A Mir, and et al. Touch probe radius compensation for coordinate measurement using kriging interpolation. Proc. Instn. Mech. Engrs., 211(Part B):11{18, 1997. [16] Yunxian Chu. Workpiece Localization: Theory, Algorithms and Implementation. PhD thesis, HKUST, March 1999. [17] H. Aoyama, M. Kawai, and T. Kishinami. A new method for detecting the contact point between a touch probe and a surface. Annals of the CIRP, 38(1):517{20, 1989. [18] W. Tiller L. Piegl. The NURBS Book. Springer-Verlag, 1st edition, 1995. [19] Y.X. Chu, J.B. Gou, and Z.X. Li. Localization algorithms: Performance evaluation and reliability analysis. In Int. Conf. on Robotics and Automation, volume 4, pages 3652{ 7, 1998. [20] R. Murray, Z.X. Li, and S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.
Table 3: Experimental results of probe radius compensation No compen. Method a Method b
Res. error(mm) Comp. Time(s) 84.2410 0.71 32.1873 2.32 11.3610 0.66
Acknowledgment The authors would like to thank Mr. G.F. Liu and Miss H. Cheng for their helpful discussion.
References
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