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Fractal Path Generation for a Metal-Mold Polishing Robot System and Its Evaluation by the Operability
Fractal Path Generation for a Metal-Mold Polishing Robot System and Its Evaluation by the Operability Yoshio Mizugaki, Masafumi Sakamoto - Submitted by Toshio Sata (1) Received on January 14,1992 This paper suggests a new method of Fractal path generation in a robot system of polishing metal molds It is conducted through the generation of a planer Peano curve in the X-Y domain and its orthogonal projection onto the free-form surface of a workpiece The orientation of the end-effector along the path is set perpendicular to the surface By calculating the sum of operability of the robotic linkage along the path the path can be evaluated from the ease of !he rcbots' motion The experiments inclusive of the polishing force control are illustrated and a brief conclusion made Keywords
Mechatronics, Polishing, Robot Programming
1 Introduction The importance of metal mold manufacturing has increased not only from the view point of industrial effects but also engineering approaches One of the most important and as yet unsolved issues is the problem of metal mold polrshing It remains from the automation of machining processes and seems to be a kind of labor consuming task The reason why it has remained outside of the automation movement is that the number of metal molds manufactured has been small and their life cycle has been relatively long The polishing process has therefore been to occupy about 30 40 O 6 of the manufacturing process Today the circumstances of metal mold manufacturing has changed The life cycle of a metal mold has shortened, a result of the variety and quick change in consumer demands in order TO develop the automatic system for a metal mold polishing process !t is essential to introduce a geometric model as a shape data of a metal mold and use an NC polishing machine Some polishing machines have been developed and reported but most of them are unaple to make a polishing path flexibly according to the shape of a metal mold and consequently vary the contact force along the path In this study the fractal path generation system has been presented as a flexible path generator based on a geometric model which is easily transformed from a solid model used in any CADCAM system Moreover the evaluation criteria of a robotic path is introduced from the view point of its operability
-
2 Polishing Robot System 2 1 System configuration The polishing robot system developed consists of an articulated industrial robot, an end-effector, a work table for setting a metal mold a robot controller and the personal computer for sensory processing essential to a Contact force control and a positioning work The system configuration is shown in Fig 1 The articulated robot has 6 degrees of freedom and its payload is 30 kg inclusive of an end-effector Its positioning repeatability is less than 0 2 mm The end-effector shown in Fig 2 consists of an air spindle motor with a grinding wheel a coiled spring for generating contact force an electric micrometer (differential transducer) for sensing a length of the spring for force control and a laser displacement meter for detecting datums of a metal mold The air spindle motor runs on the pressure of 2 5 kgf/cmz and nearly at 60000 rpm The range of contact force is thus limited to 8 through 10 N so that a fine polishing can be assumed The resolution of the laser displacement meter is 0 01 mm in the detective range of 60
mm through 140 mm Grinding wheels are those of rubber and felt buff and used with white alundum abrasive grains #800 through #3000 in machine oil The polishing path generation system presented is an associate of the CAD system of metal molds A geometric modeler is used In the path generation system and deals with a kind of lattice space model named 2-map model The other modeler in the CAD system deals with a CSG model A 2-map model represents the set of the heights of points which are obtained by the orthogonal projection in the 2-direction from the lattice points of X-Y domain onto a free-form surface Therefore a 2-map model can be easily obtained from a CSG model The data of 2-map is stored in a 2-dimensionalarray The suffix of the array indicates the X- and Y-number of a lattice point and the content of it shows its height
2 2 Fractal polishing path generation Fractal is known as a " formless" and/or "full-fledged" pattern It is discussed mathematically and some impressive examples are shown in (1.21 In this form of application. the characteristics of "plane-filling" and "scalability" of fractal curves are used which seem to be available and suitable in the most planer jobs such as polishing, gluing. painting etc The property of scalability means that of " invariant under change Of scale'' and that of "self-similarity" too A Peano curve is used as a fractal curve because its fractal dimension is 2 and satisfies the condition of "plane-filling " (Here a fractal curve is defined as a curve whose dimension is greater than 1 according to (11 The fractal dimension is defined in the following manner Let a pattern consist of B pieces of the same smaller pattern and the scale of the smaller pattern is l / A of the initial pattern Fractal dimension D can be defined as B = A' The Peano curve used in this particular application consists of 4 sub-patterns and the scale of the sub-pattern is 1/2 Therefore fractal dimension D is defined as 2 according to the formula 4=2D If a fractal curve can be limited only to a curve which has a dimension Of non-integer, a Peano curve would be out of the definition of Fractal )
device Differential transducer
Fig 1 The Configuration of a Metal-Mold Polishing Robot System
Annals of the ClRP Vol. 41/1/1992
-
Fig. 2 The Polishing End-effector
531
The procedures of fractal path generation are described below To generate a planer Peano curve in the X - Y domain of a metal mold Step 2 To map the planer Peano curves in the X - Y domain onto the surface of the workpiece A planner Peano curve is easily generated in Step 1 recursively according to the following procedures 131 Step 1-1 To specify an initiator of shape In the experiments two initiators square n and cross X are used Step 7-2 To specify the recurrent form between a pattern of order k and that of k - i In the case of square ll four basic patterns Ak Bk Ck Dk and connecting arrows are prepared Suffix k shows the order of each basic pattern Each basic pattern !n order k is represented in the following recurrent form Ak = (Dk 1. l&-arrOw Ak.1 down-arrow Ak 1 right-arrow Bk * ) Bt = (CO. i up-arrow Bk-i right-arrow Bk 1 down-arrow Ak I) CK= (BII 1 right-arrow. Ck-1, up-arrow. Cd left-arrow Dk-1) Dk = (Ak 1 , down-arrow. Dk I, left-arrow Dk up-arrow CKI ) As :he initial condition. A0 Bo Co Do are assumed as a point Thus Step 1
4
St-
1-3. To generate a pattern by means of applying Step 1-2 recursively, starting at the initiator Let's consider the number of short segments and the total length of a fractal curve Put the length of the side of the initial square as d l and that of order k as dk ( k 2 2 ) It is easily seen that there IS a constraint about the lengtk of a short segment as shown below d l = (2-1)dk (k~2j. According to the recursive process. number nk of short segments in order k k is given as 4 -1. too. Thus length hLk of the curve with order k is calculated as k k k Ah = nkdk = dl(4 -1)/(2 -1) = d1(2 + l ) . In the fractal curve adopted, it is clearly seen that the length of the curve with a current unit is about twice of that of a onelevel larger unit. The limit of JL tends towards infinity when the suffix k moves towards infinity, as shown below. k ALP Iim [Ab] lim [d1(2 +1)] = 00 ( k -->a) Namely it has no length as a fin!te and certain value This means that the fractal possesses the characteristic of a "plane-filling." As mentioned above. a square area could be filled up with a fractal curve roughly or closely by changing the scale of unit. In this application of polishing a metal mold, the scale of unit is determined according to the diameter of a grinding wheel so that the swept area of the grinding wheel covers the square area perfectly. (If the scale of a unit pattern is greater than the diameter of the grinding wheel, that is a rough fractal path, there would be some area left unpolished. On the other hand, if the unit pattern is smaller than the diameter, the whole area would be polished in an overlapped manner.) In Step 2 different methods should be prepared according to the type of geometric model used. It IS undertaken by the orthogonal projection of the planer Peano cuwe onto the free-form surface of the metal mold. In this study, lattice space model 2-map is used and because of this 2-map model structure the process in Step 2 can be performed easily There is no need for any calculation only the enumeration of elements as height in a 2-map array. While a spatial path can be generated according to the procedures mentioned above, the velocity distribution should be independently designed or generated along the path. This problem was discussed in 141. In this study a constant velocity distribution along a spatial fractal path is adopted in polishing experiments because it seems that the polishing of a metal mold might be operated under a constant velocity for the sake of fine polishing. When a free-form surface is represented as a spline surface, the mapping in Step 2 should be conducted through a calculation of the interval polynomial of the spline Now let a free-form surface be presented by Bezier surface P(u,w) of a 3rd-order polynomial. Here u and w are parameters in directions of X- and Y- axis and move in [O.l]respectively.
-
projection of a planer Peano curve made of short segments of the same length In order to generate a spatial fractal path in which the lengths of short segments are equal a planer Peano curve should be generated by taking account of the pitch of u and w parameters A deviation of parameter u in X-direction (or w in Y-direction) should be determined under the constraint of distance as follows ld(dP(u w)/du dP(u w)/du) du
= Jd(dP(u wVdw dP(u w)/dw) dw
( the domain u = [ui ui+Au ] ) ( the domain w
= [ WI w]+Aw 1 )
= Constant Generally speaking, this constraint IS so res;rictive that It is hard to implement a spatial fractal path generation system based on this conditton into a robot controller system Therefore the path generation system based on Z-map is currently programmed in this study
2 3 Path evaluation according to operability It is well known that the concept of "operability" of linkage is a useful and powerful tool for the kinematic analysis of robot linkage, positioning capability force Control and so on In case of a linkage without redundancy, namely the degree of freedom is 6. operability W of the linkage IS defined in terms of a Jacobian matrix [S]as follows, W = IdetJ I Here J is a Jacobian matrix (6 by 6) which is defined as the coefficient matrix between a spatial velocity vector and an angular velocity vector P is a generalized position vector which consists of posibon and orientation elements of the end-effector 0 is a vector of generalized variables of linkage such as joint angles dPldt=J deMt dPIdt = (dx/dt dyldt. dz/dt dddt. dpldt. dyldt). dWdt (dfJl/dt dWdt dWdt dWdt d%/dt. dWdt) Although the operability can be directly calculated according to the definition above usually it would be calculated via transformation matrix T as a product of transformation matrix A i (1sI s6) When J is divided into the 4 matrices of 3 by 3 illustrated below
-
J =
[GI
the determinant of J is given as follows det(J) = det(A). det(D-CA '6) in case of det(A)tO or det(J) = det(D). det(A-BD ' C ) in case of det(D)tO The means of evaluating a poltshing path in this study IS based on the sum of a determinant of J at both ends of the short segments of a spatial fractal path Because the sum of the determinants is directly proportional to the operability along the spatial path 3 Polishing experiments 3 1 Detection of a datum and verification of alignment work It is important to verify the positioning accuracy of the polishing robot system and the alignment work between the coordinate system of a metal mold and that of the polishing robot For the sake of the alignment work the laser displacement meter is attached at the polishing end-effector as mentioned above The alignment work is carried out by the detection of datum of a metal mold through measuring its contour lines Fig 4 shows the reliability of the laser sensor in terms of the standard deviation of measured data affected by the relative posture of the sensor to the metal mold Here Tz is an angle of posture of the laser sensor to a contour line of the metal mold The metal mold used in the polishing experiments is a cubic body with a free-form surface of convex shape Its base figure is square a normal common shape of metal molds It can easily be seen that the laser sensor is most sensitive under the condition of Tz =go" and scanning velocity s 12 5 mm/sec (75cmlmin)
yl
'fl
B2.4
B.l.4 €31.2
Bi
The elements of 4 by 4 matrix are me position vectors of control points in the Wzier surface illustrated in Fig 3 [5]For example, B,, is the position vector of control point (u.w)=(O.O). and 8 is (0.7)B, is (1.0) and B ,, is (1.1) It is important to point out that the lengths of segments of a spatial fractal curve are dtferent from each other while it is obtained through the
.
532
Dy Fig 3 A Bezier patch of a 3rd-order polynomial [5]
3 2 Contact force control [A The effect of contact force control is shown in Fig 5 as the reduction of rounding a corner edge through the comparison of a variable pressured path and a constant pressured path In the experiments of contact force control, the contact force is designed to be directed perpendicular to the free-form surface of a metal mold Firstly in cases of a constant pressured path, the contact force is kept constant until reaching the end of the corner edges of the metal mold In these instances overpolishing always occurs and a dull edge inevitably forms In contrast the contact force in a variable pressured path is designed to be inclined towards zero N to a corner edge of the mold It is clear that the latter path is useful and suitable to polishing a corner edge of metal molds
3 3 Fractal paths in polishing The polishing experiments with fractal paths have been conducted at a constant velocity distribution It is so slow e g 20 mmlsec that it seems to be no trouble in terms of self-vibrabon due to its natural frequency The fractal paths of square n and cross X patterns described in chapter 2 are shown in Fig 6 The shape of the metal mold polished is a kind of land scape and it is transformed into the model of Z-map The material of the metal mold is carbon steel S35C and the dimensional size of the enveloping cube of the metal mold is 100 mm' 100mm-100mm The comparison of surface roughness polished with fractal paths and a S-figure scanning path is illustrated in Fig 7 It is clear that fractal paths are promiscng and essential in polishing because these fractal paths consist of X-directlonalshort segments as frequently as Y- segments The ratio of numbers of X- and Y- directional short segments shown in Table 1 implies the superiority of fractal paths compared with traditional scanning paths
X l O
-3
Z0"j 190
:
..
I
0
l
5'0 7.5 100 125 150 Ve 1 oc i
t y
( c i d r n i ri)
Fig 4 The reliability of the laser displacement meter as an edge detector
3 4 Evaluation of a fractal path from the viewpoint of operability An evaluation process based on operability is produced by the following steps Step 1 To enumerate the position and orientation of the end-effector at every turning point of a spatial fractal path (The turning points respectively correspond to initial and terminal points of short segments of the planer Peano curve ) Step 2 To calculate the operability at the configuration of the robot linkage according to the position and orientation obtained in Step 1 (Firstly a calculation of the inverse kinematic problem must be performed in order to obtain the correspondingjoint angles, I e 0.from the position and orientation of the end-effector I e P Then the calculation of operability I e I det J I is carried out ) Step 3 To sum up the operability at every turning point during the Starting point and the end point of the fractal path Table 2 indicates the sum of operability according to the fractal paths at the places where the metal mold IS translated parallel to the X and Y-direction and rotated around the center of the mold The shape of the fractal path is the same as illustrated in Fig 6 By referring to the sum
Fig. 6 Fractal paths of inniators
Fig 5 The prevention of overpolishing by a variable contact force control of operability, the place of the workpiece could be numerically evaluated from the ease of the robots' motion As shown in Table 2.the position (0 600 0) in the work coordinates system and the orientation with rotated angle 0" is the most suitable configuration for polishing among the configurations simulated Nevertheless it is hard to compare the superiority between the paths passing through the same turning points because the criteria presented depends only on the combination of the points and not their permutations
I1 and X with different ranks
533
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75 105 135 165 185 Rz urn
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4 Conclusion A new method of a fractal path generation for metal mold polishing has been presented Owing to me baiaanced ratto of X- and Y-short segments of a fractal path, its superiority in polishing has been certified through the polishing experiments Moreover considering the operability along the path, the position and orientation of a workpiece could be numerically evaluated from the view of " the ease of robotic motion " Further research ISnecessary to establish a flexible path generation system with a machine intelligence which enables the system to autonomously cope with any form feature of a metal mold
Surface roughness (a)S-figure
x v
$
sacanning
Acknowledgment The authors thank Assistant Professor Keisuke Kamijo Kyushu Kyoritu University. for his discussion of a calculational procedure of fractal path generation Mr Hiroshi lkuta and Mr Satoshi Miyataka for their devoted assistance in polishing experiments and the calculation of operability
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References [l]Mandelbrot.B , 1983, The fractal geometry of nature, H.Freeman and Company, New Yo&. (The Japanese edieon is translated by Hir0naka.H.. Nikkei Science, Tokyo, 1985 ) [2]Takayasu,H.. 1986, Fractal. Asakura Printing Co , Tokyo j3lTakemura.S , 1984, Graphic Art by Personal Computer (2nd ed),VAN researchlohm-sha. Kyoto. [4]Mizugaki.Y.. Kirnura.F .Sata.T ,Suzuki.T ,1984.Generation of a free curve trajectory with a specified velocity distribution for an articulated robot, Proc Int'l Symposium on Design and Synthesis, 691-696. [5]Rogers. D F , Adams.J A , 1976, Mathematical Elements for Computer Graphics. McGraw-Hill. USA. (Its Japanese edition by Yamaguchi.F.. 1979, Nikkan Kogyo. Tokyo.) [G]Yoshikawa.T.,1984, Analysis and Control of Robot Manipulators with Redundancy, Robotics Research, The First International Symposium. eds. 6rady.M. and Paul.R., The MIT Press, pp.735-747 [7]Mizugaki.Y., Sakamoto.M., Kamij0.K / Taniguchi.N.(l). 1990, OeveloDment of a Metal Mold Polishina Robot Svstem with Contact Pressure Control Using CAD/CAM data. Annais of the ClRP VoI. 3911I1 990, 523-526 [8]Dictionary of Mathematics, 1989, lwanami Press, Tokyo
[m
Fig 7 Frequency distribution of surface roughness of a polished plane Table 1 The ratio of X- and Y- short segments of polishing paths The number of short vector!
Tool
P a t h
t
i
1
+
*
SUM
S-figure scanning pattern
Rotated angle around the center of the workpiece Position placed in the work coordinates system (X.Y.Z).
0"
90"
180" 270"
(X.Y.Z) = (0, 0 , O )
0.44
0.08
'0.11
0.07
(X,Y,Z)=( 200, 0, 0)
0.28
0.20
0.33
0.19
= (-200, 0. 0)
0.42
0.07
0.16
0.09
(X,Y.Z)=(0,600,0)
0.53
0.18
0.13
0.16
(X.Y,Z)=( 0, -600,O)
0.47
0.03
0.04
0.03
(X.Y.z)
Peano pattern of
n
Peano pattern o f
X
1 2 3
(52%)
534
(48%)
(100%)
Mechatronics, Polishing, Robot Programming
1 Introduction The importance of metal mold manufacturing has increased not only from the view point of industrial effects but also engineering approaches One of the most important and as yet unsolved issues is the problem of metal mold polrshing It remains from the automation of machining processes and seems to be a kind of labor consuming task The reason why it has remained outside of the automation movement is that the number of metal molds manufactured has been small and their life cycle has been relatively long The polishing process has therefore been to occupy about 30 40 O 6 of the manufacturing process Today the circumstances of metal mold manufacturing has changed The life cycle of a metal mold has shortened, a result of the variety and quick change in consumer demands in order TO develop the automatic system for a metal mold polishing process !t is essential to introduce a geometric model as a shape data of a metal mold and use an NC polishing machine Some polishing machines have been developed and reported but most of them are unaple to make a polishing path flexibly according to the shape of a metal mold and consequently vary the contact force along the path In this study the fractal path generation system has been presented as a flexible path generator based on a geometric model which is easily transformed from a solid model used in any CADCAM system Moreover the evaluation criteria of a robotic path is introduced from the view point of its operability
-
2 Polishing Robot System 2 1 System configuration The polishing robot system developed consists of an articulated industrial robot, an end-effector, a work table for setting a metal mold a robot controller and the personal computer for sensory processing essential to a Contact force control and a positioning work The system configuration is shown in Fig 1 The articulated robot has 6 degrees of freedom and its payload is 30 kg inclusive of an end-effector Its positioning repeatability is less than 0 2 mm The end-effector shown in Fig 2 consists of an air spindle motor with a grinding wheel a coiled spring for generating contact force an electric micrometer (differential transducer) for sensing a length of the spring for force control and a laser displacement meter for detecting datums of a metal mold The air spindle motor runs on the pressure of 2 5 kgf/cmz and nearly at 60000 rpm The range of contact force is thus limited to 8 through 10 N so that a fine polishing can be assumed The resolution of the laser displacement meter is 0 01 mm in the detective range of 60
mm through 140 mm Grinding wheels are those of rubber and felt buff and used with white alundum abrasive grains #800 through #3000 in machine oil The polishing path generation system presented is an associate of the CAD system of metal molds A geometric modeler is used In the path generation system and deals with a kind of lattice space model named 2-map model The other modeler in the CAD system deals with a CSG model A 2-map model represents the set of the heights of points which are obtained by the orthogonal projection in the 2-direction from the lattice points of X-Y domain onto a free-form surface Therefore a 2-map model can be easily obtained from a CSG model The data of 2-map is stored in a 2-dimensionalarray The suffix of the array indicates the X- and Y-number of a lattice point and the content of it shows its height
2 2 Fractal polishing path generation Fractal is known as a " formless" and/or "full-fledged" pattern It is discussed mathematically and some impressive examples are shown in (1.21 In this form of application. the characteristics of "plane-filling" and "scalability" of fractal curves are used which seem to be available and suitable in the most planer jobs such as polishing, gluing. painting etc The property of scalability means that of " invariant under change Of scale'' and that of "self-similarity" too A Peano curve is used as a fractal curve because its fractal dimension is 2 and satisfies the condition of "plane-filling " (Here a fractal curve is defined as a curve whose dimension is greater than 1 according to (11 The fractal dimension is defined in the following manner Let a pattern consist of B pieces of the same smaller pattern and the scale of the smaller pattern is l / A of the initial pattern Fractal dimension D can be defined as B = A' The Peano curve used in this particular application consists of 4 sub-patterns and the scale of the sub-pattern is 1/2 Therefore fractal dimension D is defined as 2 according to the formula 4=2D If a fractal curve can be limited only to a curve which has a dimension Of non-integer, a Peano curve would be out of the definition of Fractal )
device Differential transducer
Fig 1 The Configuration of a Metal-Mold Polishing Robot System
Annals of the ClRP Vol. 41/1/1992
-
Fig. 2 The Polishing End-effector
531
The procedures of fractal path generation are described below To generate a planer Peano curve in the X - Y domain of a metal mold Step 2 To map the planer Peano curves in the X - Y domain onto the surface of the workpiece A planner Peano curve is easily generated in Step 1 recursively according to the following procedures 131 Step 1-1 To specify an initiator of shape In the experiments two initiators square n and cross X are used Step 7-2 To specify the recurrent form between a pattern of order k and that of k - i In the case of square ll four basic patterns Ak Bk Ck Dk and connecting arrows are prepared Suffix k shows the order of each basic pattern Each basic pattern !n order k is represented in the following recurrent form Ak = (Dk 1. l&-arrOw Ak.1 down-arrow Ak 1 right-arrow Bk * ) Bt = (CO. i up-arrow Bk-i right-arrow Bk 1 down-arrow Ak I) CK= (BII 1 right-arrow. Ck-1, up-arrow. Cd left-arrow Dk-1) Dk = (Ak 1 , down-arrow. Dk I, left-arrow Dk up-arrow CKI ) As :he initial condition. A0 Bo Co Do are assumed as a point Thus Step 1
4
St-
1-3. To generate a pattern by means of applying Step 1-2 recursively, starting at the initiator Let's consider the number of short segments and the total length of a fractal curve Put the length of the side of the initial square as d l and that of order k as dk ( k 2 2 ) It is easily seen that there IS a constraint about the lengtk of a short segment as shown below d l = (2-1)dk (k~2j. According to the recursive process. number nk of short segments in order k k is given as 4 -1. too. Thus length hLk of the curve with order k is calculated as k k k Ah = nkdk = dl(4 -1)/(2 -1) = d1(2 + l ) . In the fractal curve adopted, it is clearly seen that the length of the curve with a current unit is about twice of that of a onelevel larger unit. The limit of JL tends towards infinity when the suffix k moves towards infinity, as shown below. k ALP Iim [Ab] lim [d1(2 +1)] = 00 ( k -->a) Namely it has no length as a fin!te and certain value This means that the fractal possesses the characteristic of a "plane-filling." As mentioned above. a square area could be filled up with a fractal curve roughly or closely by changing the scale of unit. In this application of polishing a metal mold, the scale of unit is determined according to the diameter of a grinding wheel so that the swept area of the grinding wheel covers the square area perfectly. (If the scale of a unit pattern is greater than the diameter of the grinding wheel, that is a rough fractal path, there would be some area left unpolished. On the other hand, if the unit pattern is smaller than the diameter, the whole area would be polished in an overlapped manner.) In Step 2 different methods should be prepared according to the type of geometric model used. It IS undertaken by the orthogonal projection of the planer Peano cuwe onto the free-form surface of the metal mold. In this study, lattice space model 2-map is used and because of this 2-map model structure the process in Step 2 can be performed easily There is no need for any calculation only the enumeration of elements as height in a 2-map array. While a spatial path can be generated according to the procedures mentioned above, the velocity distribution should be independently designed or generated along the path. This problem was discussed in 141. In this study a constant velocity distribution along a spatial fractal path is adopted in polishing experiments because it seems that the polishing of a metal mold might be operated under a constant velocity for the sake of fine polishing. When a free-form surface is represented as a spline surface, the mapping in Step 2 should be conducted through a calculation of the interval polynomial of the spline Now let a free-form surface be presented by Bezier surface P(u,w) of a 3rd-order polynomial. Here u and w are parameters in directions of X- and Y- axis and move in [O.l]respectively.
-
projection of a planer Peano curve made of short segments of the same length In order to generate a spatial fractal path in which the lengths of short segments are equal a planer Peano curve should be generated by taking account of the pitch of u and w parameters A deviation of parameter u in X-direction (or w in Y-direction) should be determined under the constraint of distance as follows ld(dP(u w)/du dP(u w)/du) du
= Jd(dP(u wVdw dP(u w)/dw) dw
( the domain u = [ui ui+Au ] ) ( the domain w
= [ WI w]+Aw 1 )
= Constant Generally speaking, this constraint IS so res;rictive that It is hard to implement a spatial fractal path generation system based on this conditton into a robot controller system Therefore the path generation system based on Z-map is currently programmed in this study
2 3 Path evaluation according to operability It is well known that the concept of "operability" of linkage is a useful and powerful tool for the kinematic analysis of robot linkage, positioning capability force Control and so on In case of a linkage without redundancy, namely the degree of freedom is 6. operability W of the linkage IS defined in terms of a Jacobian matrix [S]as follows, W = IdetJ I Here J is a Jacobian matrix (6 by 6) which is defined as the coefficient matrix between a spatial velocity vector and an angular velocity vector P is a generalized position vector which consists of posibon and orientation elements of the end-effector 0 is a vector of generalized variables of linkage such as joint angles dPldt=J deMt dPIdt = (dx/dt dyldt. dz/dt dddt. dpldt. dyldt). dWdt (dfJl/dt dWdt dWdt dWdt d%/dt. dWdt) Although the operability can be directly calculated according to the definition above usually it would be calculated via transformation matrix T as a product of transformation matrix A i (1sI s6) When J is divided into the 4 matrices of 3 by 3 illustrated below
-
J =
[GI
the determinant of J is given as follows det(J) = det(A). det(D-CA '6) in case of det(A)tO or det(J) = det(D). det(A-BD ' C ) in case of det(D)tO The means of evaluating a poltshing path in this study IS based on the sum of a determinant of J at both ends of the short segments of a spatial fractal path Because the sum of the determinants is directly proportional to the operability along the spatial path 3 Polishing experiments 3 1 Detection of a datum and verification of alignment work It is important to verify the positioning accuracy of the polishing robot system and the alignment work between the coordinate system of a metal mold and that of the polishing robot For the sake of the alignment work the laser displacement meter is attached at the polishing end-effector as mentioned above The alignment work is carried out by the detection of datum of a metal mold through measuring its contour lines Fig 4 shows the reliability of the laser sensor in terms of the standard deviation of measured data affected by the relative posture of the sensor to the metal mold Here Tz is an angle of posture of the laser sensor to a contour line of the metal mold The metal mold used in the polishing experiments is a cubic body with a free-form surface of convex shape Its base figure is square a normal common shape of metal molds It can easily be seen that the laser sensor is most sensitive under the condition of Tz =go" and scanning velocity s 12 5 mm/sec (75cmlmin)
yl
'fl
B2.4
B.l.4 €31.2
Bi
The elements of 4 by 4 matrix are me position vectors of control points in the Wzier surface illustrated in Fig 3 [5]For example, B,, is the position vector of control point (u.w)=(O.O). and 8 is (0.7)B, is (1.0) and B ,, is (1.1) It is important to point out that the lengths of segments of a spatial fractal curve are dtferent from each other while it is obtained through the
.
532
Dy Fig 3 A Bezier patch of a 3rd-order polynomial [5]
3 2 Contact force control [A The effect of contact force control is shown in Fig 5 as the reduction of rounding a corner edge through the comparison of a variable pressured path and a constant pressured path In the experiments of contact force control, the contact force is designed to be directed perpendicular to the free-form surface of a metal mold Firstly in cases of a constant pressured path, the contact force is kept constant until reaching the end of the corner edges of the metal mold In these instances overpolishing always occurs and a dull edge inevitably forms In contrast the contact force in a variable pressured path is designed to be inclined towards zero N to a corner edge of the mold It is clear that the latter path is useful and suitable to polishing a corner edge of metal molds
3 3 Fractal paths in polishing The polishing experiments with fractal paths have been conducted at a constant velocity distribution It is so slow e g 20 mmlsec that it seems to be no trouble in terms of self-vibrabon due to its natural frequency The fractal paths of square n and cross X patterns described in chapter 2 are shown in Fig 6 The shape of the metal mold polished is a kind of land scape and it is transformed into the model of Z-map The material of the metal mold is carbon steel S35C and the dimensional size of the enveloping cube of the metal mold is 100 mm' 100mm-100mm The comparison of surface roughness polished with fractal paths and a S-figure scanning path is illustrated in Fig 7 It is clear that fractal paths are promiscng and essential in polishing because these fractal paths consist of X-directlonalshort segments as frequently as Y- segments The ratio of numbers of X- and Y- directional short segments shown in Table 1 implies the superiority of fractal paths compared with traditional scanning paths
X l O
-3
Z0"j 190
:
..
I
0
l
5'0 7.5 100 125 150 Ve 1 oc i
t y
( c i d r n i ri)
Fig 4 The reliability of the laser displacement meter as an edge detector
3 4 Evaluation of a fractal path from the viewpoint of operability An evaluation process based on operability is produced by the following steps Step 1 To enumerate the position and orientation of the end-effector at every turning point of a spatial fractal path (The turning points respectively correspond to initial and terminal points of short segments of the planer Peano curve ) Step 2 To calculate the operability at the configuration of the robot linkage according to the position and orientation obtained in Step 1 (Firstly a calculation of the inverse kinematic problem must be performed in order to obtain the correspondingjoint angles, I e 0.from the position and orientation of the end-effector I e P Then the calculation of operability I e I det J I is carried out ) Step 3 To sum up the operability at every turning point during the Starting point and the end point of the fractal path Table 2 indicates the sum of operability according to the fractal paths at the places where the metal mold IS translated parallel to the X and Y-direction and rotated around the center of the mold The shape of the fractal path is the same as illustrated in Fig 6 By referring to the sum
Fig. 6 Fractal paths of inniators
Fig 5 The prevention of overpolishing by a variable contact force control of operability, the place of the workpiece could be numerically evaluated from the ease of the robots' motion As shown in Table 2.the position (0 600 0) in the work coordinates system and the orientation with rotated angle 0" is the most suitable configuration for polishing among the configurations simulated Nevertheless it is hard to compare the superiority between the paths passing through the same turning points because the criteria presented depends only on the combination of the points and not their permutations
I1 and X with different ranks
533
15
45
75 105 135 165 185 Rz urn
XlO-'
4 Conclusion A new method of a fractal path generation for metal mold polishing has been presented Owing to me baiaanced ratto of X- and Y-short segments of a fractal path, its superiority in polishing has been certified through the polishing experiments Moreover considering the operability along the path, the position and orientation of a workpiece could be numerically evaluated from the view of " the ease of robotic motion " Further research ISnecessary to establish a flexible path generation system with a machine intelligence which enables the system to autonomously cope with any form feature of a metal mold
Surface roughness (a)S-figure
x v
$
sacanning
Acknowledgment The authors thank Assistant Professor Keisuke Kamijo Kyushu Kyoritu University. for his discussion of a calculational procedure of fractal path generation Mr Hiroshi lkuta and Mr Satoshi Miyataka for their devoted assistance in polishing experiments and the calculation of operability
pattern
6o
50 40 30
:2 0 3
cr. 1 0 0
15
45
Surface
75 105 135 165 195 roughness
(b)P e a n o p a t t e r n x v
1
Rz [Ill
XlO-'
fim
60
50 40 30 20
;
cr. 10 0 15
45
Surface
75 105 135 165 185 roughness
(c) Peano p a t t e r n
XlO-'
Rz urn
References [l]Mandelbrot.B , 1983, The fractal geometry of nature, H.Freeman and Company, New Yo&. (The Japanese edieon is translated by Hir0naka.H.. Nikkei Science, Tokyo, 1985 ) [2]Takayasu,H.. 1986, Fractal. Asakura Printing Co , Tokyo j3lTakemura.S , 1984, Graphic Art by Personal Computer (2nd ed),VAN researchlohm-sha. Kyoto. [4]Mizugaki.Y.. Kirnura.F .Sata.T ,Suzuki.T ,1984.Generation of a free curve trajectory with a specified velocity distribution for an articulated robot, Proc Int'l Symposium on Design and Synthesis, 691-696. [5]Rogers. D F , Adams.J A , 1976, Mathematical Elements for Computer Graphics. McGraw-Hill. USA. (Its Japanese edition by Yamaguchi.F.. 1979, Nikkan Kogyo. Tokyo.) [G]Yoshikawa.T.,1984, Analysis and Control of Robot Manipulators with Redundancy, Robotics Research, The First International Symposium. eds. 6rady.M. and Paul.R., The MIT Press, pp.735-747 [7]Mizugaki.Y., Sakamoto.M., Kamij0.K / Taniguchi.N.(l). 1990, OeveloDment of a Metal Mold Polishina Robot Svstem with Contact Pressure Control Using CAD/CAM data. Annais of the ClRP VoI. 3911I1 990, 523-526 [8]Dictionary of Mathematics, 1989, lwanami Press, Tokyo
[m
Fig 7 Frequency distribution of surface roughness of a polished plane Table 1 The ratio of X- and Y- short segments of polishing paths The number of short vector!
Tool
P a t h
t
i
1
+
*
SUM
S-figure scanning pattern
Rotated angle around the center of the workpiece Position placed in the work coordinates system (X.Y.Z).
0"
90"
180" 270"
(X.Y.Z) = (0, 0 , O )
0.44
0.08
'0.11
0.07
(X,Y,Z)=( 200, 0, 0)
0.28
0.20
0.33
0.19
= (-200, 0. 0)
0.42
0.07
0.16
0.09
(X,Y.Z)=(0,600,0)
0.53
0.18
0.13
0.16
(X.Y,Z)=( 0, -600,O)
0.47
0.03
0.04
0.03
(X.Y.z)
Peano pattern of
n
Peano pattern o f
X
1 2 3
(52%)
534
(48%)
(100%)