Rapid Prototyping Model Coordinate Estimation Using Radial Basis Function

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					                                             (IJCSIS) International Journal of Computer Science and Information Security,
                                             Vol. 9, No. 2, February 2011




      Rapid Prototyping Model Coordinate
     Estimation Using Radial Basis Function.
                               1
                                Anantmurty S. Shastry and 2S.Purushothaman
           1
               Anantmurty S. Shastry
                                                                   2
                 Research Scholar,                                     Dr.S.Purushothaman, Principal ,
    Department of Mechanical Engineering                       Sun College of Engineering and Technology,
         Vinayaka Missions University,                                       Sun Nagar, Erachakulum,
            Salem, Tamilnadu, India                                    Kanyakumari district-629902,India
        E-Mail: ansshastry@yahoo.co.in                           E-Mail: dr.s.purushothaman@gmail.com
 
ABSTRACT: This paper discusses the methods for              making this decision, the designers and RP
getting proper geometric coordinates of a sample            machine operators should consider a number of
object that has to be rapid prototyped. The                 different processes and specific constraints. This
coordinates of the objects is obtained by using             may be a difficult and time consuming task.
Radial Basis Function (RBF). The training is done
with many sample objects. It is expected to have                      The RP material flows through an
minimum distance traveled by the Rapid                      orifice and comes out in the form of drops. The
prototyping machine when the software follows the           size of the drop is depending upon the speed of
geometric coordinates produced by the RBF.                  the wire comes out and solidification of material.
                                                            For example, 1 mm size of drop is placed in 1
           Key words: Rapid Prototyping, Artificial         mm size cube cavity to get the same size of cube
Neural Network, Radial Basis Function.                      after solidification in fraction of seconds[1]. The
                                                            sides of the cube should be flat in all respects. To
                                                            achieve this focus has been made on a method
                                                            which can inform that how to make the above
                                                            things with critical path method (CPM)[6]. Some
                 1.   INTRODUCTION                          products have been chosen with their
                                                            applications, particularly in medical area. By
                                                            considering all the parameters in developing any
     Rapid prototyping (RP) refers to a variety of          kind of object is being able to produce in shorter
specialized equipment, software and materials               time without any difficulty[3].
capable of using 3D computer aided design
(CAD)[5] data input to directly fabricate                               2.   MATERIALS AND METHODS
geometrically complex objects. RP technologies
have emerged as a key element of time with their
ability to shorten the product design and                        2.1 Materials
development process[2]. This highly innovative
and cost efficient technology has found                         A schematic flow of the proposed work is
applications in automotive, aerospace and                   presented in Figure 1.
medical equipment manufacturing, replacing the                  Rapid Model: It is the end product that has to
commonly used slower and less accurate manual               be rapid prototyped.
methods of fabricating prototypes[4]. With
advances in established technologies, materials                Coordinates: There are various Coordinates
and the introduction of new methods, selecting              measured from the RP model either through
the right RP machine has become much more                   CMM/Reverse Engineering/existing drawing
difficult and is one of the most important                  details.
decisions to be made when employing any RP
technology. This is vital in minimizing built
time, cost and achieving optimal accuracy. When



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                                                                                        ISSN 1947-5500
                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                              Vol. 9, No. 2, February 2011




   Sizes: The length, width /thickness,                          where
breadth/height and other profiles are calculated
from the coordinates.                                                          c is a vector containing the
                                                                            coefficients of the RBF,
   RBF: Coordinates and sizes of sample RP
models are used as data for training the RBF                                   R is a vector containing the
neural network to obtain final weights that will                            centres of the RBF, and
be used for testing.                                                            φ is the basis function or
   Obtain format to meet RP M/c: The outputs                                activation function of the network.
of RBF are used as inputs for RP M/c converter                   Implementation
where RP model will be developed.
                                                                 Step 1: Apply Radial Basis Function.
     Rp Model                 Co‐ordinates
                                                                             No. of Input = 15
                                                                             No. of Patterns = 6
                                                                             No. of Centre = 6
                                                                           Calculate RBF as
     Input to RBF             Define sizes 
                                                                            RBF = exp (-X)
                                                                        Calculate Matrix as
                                                                            G = RBF
                                                                            A = GT * G
      Obtain format to                                                 Calculate
      meet RP M/c                                                           B = A-1
                                                                       Calculate
            Fig.1 Schematic flow
                                                                            E = B * GT
                                                                 Step 2: Calculate the Final Weight.
    2.2 Methods                                                             F=E*D
                                                                 Step 3: Store the Final Weights in a File.

            The concept of distance measure is                        3.         EXPERIMENT SET UP
used to associate the input and output pattern
values. Radial Basis Functions is capable of                     Six RP models have been considered as
producing approximations to an unknown                       examples for testing the RBF network. Each RP
function ‘f’ from a set of input data abscissa. The          model has been labeled with Cartesian
approximation is produced by passing an input                coordinates. The models have been developed
point through a set of basis functions, each of              using CAD software. The models are defined
which contains one of the RBF centres,                       with definite number of points. The distance
multiplying the result of each function by a                 between points are calculated internally by the
coefficient and then summing them linearly.                  program. During training RBF, only the point
            For     each    function     ‘t’,  the           coordinates are input in the input layer. The
approximation to this function is essentially                number of centers used is 6. The targets used is
stored in the coefficients and centres of the RBF.           15.
These parameters are in no way unique, since for
each function ‘t’ being approximated, many                      Table 1 presents 6 sample RP models under
combinations of parameter values exist. RBFs                 consideration. Table 2 presents number of points
have the following mathematical representation:              considered in this analysis for each RP model.
              N −1
                                                             Table 3a-c presents actual coordinates in mm for
F(x) = c o + ∑ c i Φ(|| x − R i ||)
                                                             each point. The total number of points
                                              (1)            considered is 15 in each object.
              i =0




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                                                                                         ISSN 1947-5500
                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                          Vol. 9, No. 2, February 2011




                                                         Table 1 Sample RP models




Table 2 Number of points in the RP model
RP                      Number of points
1                       8
2                       10
3                       12
4                       15
5                       4
6                       5

                                                     Table 3a Cartesian coordinate
               P1                       P2                    P3                          P4                            P5
       x            y        z     x         y   z   x             y         z    x            y        z       x            y       z
1          0        0        0       50      0   0      50            50     0        0           50    50          0            0   50
2       9.08        0        0    38.47      0   0   47.55         27.95     0    23.77        45.22     0          0        27.95    0

3       12.5        0        0     37.5      0   0       50        21.65     0     37.5        43.30    0        12.5        43.30    0

4      13.52        0        0    32.66      0   0   46.19         13.52     0    46.19        32.66    0      32.66         46.19    0

5          0        0        0         25    0   0    12.5         21.65     0     12.5         7.22    50          x            x    x

6          0        0        0         25    0   0       25            25    0        0            25   0        12.5         7.22   50



           4        RESULTS AND DISCUSSION
                                                                             the actual implementation, the RP model
     The coordinates of the RP models are learnt                             coordinates are given as inputs to the RBF to
by RBF. Table 4 presents the outputs of RBF for                              obtain the actual coordinates that helps in RP
all the 6 RP models for the points p1, p2. Similar                           modeling.
close outputs are obtained for points p3, p4, p5,
p6, p7, p8, p9, p10, p11, p12, p13, p14, p15
    Conclusion: This work has made an attempt
to train RBF with RP model coordinates. During




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                                                                                                         ISSN 1947-5500
                                                                                    (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                    Vol. 9, No. 2, February 2011




                                                                           Table 3b Cartesian coordinate
               P6                                        P7                                        P8                                     P9                                       P10
     x              y          z              x               y                z          x              y             z         x             y                 z         x         y              z
1         50            0   50                     50         50           50                 0              50    50                x                 x         x             x             x      x

2    9.08               0   50            38.47                   0        50            47.55          27.95      50           23.77         45.22           50               0       27.95       50

3         0     21.65          0              12.5                0        50             37.5               0     50                50       21.65           50           37.5        43.30       50

4   13.52       46.19          0                   0     32.66                 0              0         13.52          0        13.52                  0      50          32.66              0     50

5         x             x      x                   x              x            x              x              x         x             x                 x         x             x             x      x

6         x             x      x                   x              x            x              x              x         x             x                 x         x             x             x      x


 

                                                                               Table 3c Cartesian coordinate

               P11                                        P12                                            P13                                   P14                                     P15
     x            y                z           x             y                      z         x                y           z          x                 y             z        x        y           z
1          x            x           x               x                      x         x             x               x        x    x                 x                 x         x   x               x

2          x            x          x                x                      x         x             x               x        x    x                 x                 x         x   x               x

3    12.5          43.30       50                   0             21.65            50              x               x       x     x                 x                 x         x   x               x

4   46.19          13.52       50             46.19               32.66            50         32.66          46.19         50        13.52             46.19         50        0       32.66       50

5          x            x          x                x                      x         x             x               x        x             x                  x        x        x   x               x

6          x            x          x                x                      x         x             x               x        x             x                  x        x        x             x      x

X   represents no coordinates




                        REFERENCES                                                                                 [4]      Pham, D.T., and Pham, P.T. N. Computational

    [1]    Rao, P.N., Lerner, Y. and Kouznetsov, V. Rapid                                                                   Intelligence for Manufacturing. Computational

           Prototyping              Applications                      in             Metal                                  Intelligence in Manufacturing Handbook, CRC

           Casting.Institution                of        Engineers                  Journal,                                 Press, New York, 2000.

           Malaysia. Vol. 64, No.3, 2003, pp.1-7.                                                                  [5]      Fadel, G.M. and Kirschman, C. “ Accuracy Issues

    [2]    Pham,        D.T.            and        Dimov,             S.S.          Rapid                                   in CAD to RP Translations”, Invited paper to the

           Manufacturing: The Technologies & Applications                                                                   first Internet conference on Rapid Prototyping,

           of Rapid Prototyping & Rapid Tooling. Springer-                                                                  Forwarded to Rapid Prototyping Journal, 1995

           Verlag, London, 2001.Proceedings of The 2006                                                            [6]      Wodziak, J. R., Fadel, G. M. and Kirschman, C.

           IJME - INTERTECH Conference                                                                                      F., “A Genetic Algorithm for Optimizing multiple

    [3]    Fadel, G.M., and Ganti, R. ”Parametric Based                                                                     part placement to reduce build time”, Paper

           Controller For Rapid Prototyping Applications”                                                                   presented     at       the        Fifth        International         Rapid

           Presented at the 1998 Solid Freeform Fabrication                                                                 Prototyping Conference, Dayton OH, 1994,

           Conference, Austin, TX, 1998                                                                                     published in the conference proceedings.




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                                                                                                                                                            ISSN 1947-5500
                                                                                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                         Vol. 9, No. 2, February 2011




                                                                                                                          Table 4 RBF outputs

                                                                             X -coordinate of P1                                                                                                                                       Y -coordinate of P1
                                          14                                                                                                                                                           1
                                                                                                                     Target                                                                                                                                                      Target
                                                                                                                     RBF output                                                                   0.8                                                                            RBF output
                                          12




                                                                                                                                                   Estimated and target coordinate outputs
Estimated and target coordinate outputs




                                                                                                                                                                                                  0.6

                                          10                                                                                                                                                      0.4

                                                                                                                                                                                                  0.2
                                           8
                                                                                                                                                                                                       0
                                           6
                                                                                                                                                                                                  -0.2


                                           4                                                                                                                                                      -0.4

                                                                                                                                                                                                  -0.6
                                           2
                                                                                                                                                                                                  -0.8

                                           0                                                                                                                                                       -1
                                               1       1.5    2       2.5     3          3.5      4       4.5    5          5.5       6                                                                    1     1.5       2    2.5        3      3.5         4    4.5       5       5.5          6
                                                                                      RP model                                                                                                                                                 RP model




                                                                              Z -coordinate of P1                                                                                                                                     X -coordinate of P2
                                               1                                                                                                                                                  55
                                                                                                                         Target                                                                                                                                              Target
                                          0.8                                                                            RBF output                                                                                                                                          RBF output
                                                                                                                                                                                                  50
Estimated and target coordinate outputs




                                                                                                                                              Estimated and target coordinate outputs



                                          0.6

                                          0.4                                                                                                                                                     45

                                          0.2
                                                                                                                                                                                                  40
                                               0
                                                                                                                                                                                                  35
                                          -0.2

                                          -0.4
                                                                                                                                                                                                  30

                                          -0.6
                                                                                                                                                                                                  25
                                          -0.8

                                           -1                                                                                                                                                     20
                                                   1    1.5       2    2.5        3       3.5         4    4.5       5       5.5          6                                                            1        1.5    2       2.5     3      3.5         4       4.5    5         5.5        6
                                                                                       RP model                                                                                                                                            RP model


                                                                              Y -coordinate of P2                                                                                                                                       Z -coordinate of P2
                                               1                                                                                                                                                       1
                                                                                                                         Target                                                                                                                                                  Target
                                          0.8                                                                            RBF output                                                                0.8                                                                           RBF output
Estimated and target coordinate outputs




                                                                                                                                                        Estimated and target coordinate outputs




                                          0.6                                                                                                                                                      0.6

                                          0.4                                                                                                                                                      0.4

                                          0.2                                                                                                                                                      0.2

                                               0                                                                                                                                                       0

                                          -0.2                                                                                                                                                    -0.2

                                          -0.4                                                                                                                                                    -0.4

                                          -0.6                                                                                                                                                    -0.6

                                          -0.8                                                                                                                                                    -0.8

                                           -1                                                                                                                                                          -1
                                                   1    1.5       2    2.5        3       3.5         4    4.5       5       5.5          6                                                                 1    1.5       2    2.5        3      3.5         4    4.5       5       5.5          6
                                                                                       RP model                                                                                                                                                RP model




  




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