神兜兜和他的朋友们

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A. Caballero-Ruiz et al. / Mechatronics 17 (2007) 231–243 239
x
2 2 The chromosomes population is randomly created and is
dx 1/4 Cxx1X tCyxY tCzxZ tCxx2X sin t/x
p evaluated by an aptitude function defined by
rpp
mq?????????????????????????????????????????????????????????????????????????????? 2
tY cos tayx tZcos tazx ; 0 X 0 2 0 2 0 2
2 2 erri 1/4 eXliX0T teYliY0T teZliZ0T eRtrT
l1/41
y
2 2
dy 1/4 Cyy1Y tCxyX tCzyZ tCyy2Y sin t/y
16
p e T
r
e13T
0pp X 1/4 Xl tdxi;
li
tX cos taxy tZcos tazy ;
2 2 0 Yli 1/4 Yl tdyi; l 1/4 1;2;3;...;m; i 1/4 1;2;3;...;n e17T
z
2 2 0
dz 1/4 Czz1Z tCxzX tCyzY tCzz2Zsin t/z Zli 1/4 Zl tdzi;
p
rpp where erri is the quadratic error of the modified experimen-
tX cos taxz tY cos tayz ; 0 0 0
2 2 tal data Xli;Yli;Zli that is obtained by means of m experi-
mental data Xl, Yl and Zl a?ected by the error deviations
here the constants Cxx1, Cyy1, Czz1, Cxx2, Cyy2, Czz2, /x,
dx, dy and dz calculated from a population of ny and /z represent the translational errors; Cxx1, Cyy1
chromosomes.
nd Czz1 are the position error of the corresponding axis,
The GA generates a population of n chromosomes,
hich can be associated with mechanical parts such as
whose aptitude function is calculated. After that, the 20%
he gearboxes, the detection of home position, the straight-
of the population is considered as the fathers of the next
essoftheaxis,etc.andCxx2,Cyy2,Czz2,/x,/y and/z rep-
generation. A mutation is applied to the fathers to generate
esent the positional error corresponding to the produced
a new population of chromosomes, while the center of the
rrors by non constant thread in the lead screw and Pr is
sphere is recalculated taking into account the best chromo-
he lead of the lead screws. The constants Cyx, Czx, Cxy,
some of the current generation. The algorithm realizes azy, Cxz and Cyz represent how the straightness of comple-
defined number of generations to find the chromosome
mentary axes modify the errors the corresponding axis; in
set that presents the best fit for the aptitude function and
his case, this error can include pitch and yaw errors.Final-
selects it as thesolution ofthe problem.Therefore, thecon-
y, the constants ayx, azx, axy, azy, axz and ayz represent the
stants C , C , C , C , / , a , a , C , C , C , C , / ,
xx1 xx2 yx zx x yx zx yy1 yy2 xy zy y
eviation angles between the axes. The equations were pro-
a ,a ,C ,C ,C ,C ,/ ,a ,a canbeemployedinthe
xy zy zz1 zz2 xz yz z xz yz
osed after long time of study and experimentation with
deviations model to develop a control system to improve
he MMT presented in this paper.
the MMT function capabilities.
The experimental data are modified using the following
quation:
5.2. Experimental results
0
X 1/4 X tdx;
0
Y 1/4 Y tdy; e14T The proposed algorithms were experimentally tested
0 using two 5/16 in. (7.937 mm) stainless steel balls with dia-
Z 1/4 Z tdz;
metrical deviations lower than 0.6 lm. The balls are fixed
0 0 0
here X , Y , Z are the modified experimental data; X, Y, to shanks by epoxy glue and the electrical contact is pro- are the experimental data; and dx, dy, dz are the error duced by a wire that joins the ball and the shank. In the
eviation in the MMT. metrology process, a group of 233 measurements were con-
In order to determine the constants included in Eq. (13), sidered as the experimental test database for each B2 posi-
e use a similar algorithm to the algorithm used to find the tion. For the test of the algorithms, measurements for 70
enter of B2. In this case, the chromosome population was di?erent positions of B2 were made. These measurements
omposed by groups that contain all the constants pre- was obtained by rotating the B2 support using the spindle
ented in Eq. (14) as is shown in Eq. (15): along 14 di?erent positions and by moving the B2 support
along the Y axis in increments of 2 mm starting fromxx11;Cxx21;Cyx1;Czx1;/x1;ayx1;azx1;Cyy11;Cyy21;Cxy1;
11 mm from the spindle until 19 mm.zy1;/y1;axy1;azy1;Czz11;Czz21;Cxz1;Cyz1;/z1;axz1;ayz1:
In order to test the developed GA to calculate the centerchromosome 1; of B2, the best results were obtained with a chromosomexx12;Cxx22;Cyx2;Czx2;/x2;ayx2;azx2;Cyy12;Cyy22;Cxy2; population of 500 entities during 100 generations and the
mutation employed to create each next generation was dis-zy2;/y2;axy2;azy2;Czz12;Czz22;Cxz2;Cyz2;/z2;axz2;ayz2:
tributed in di?erent range of values as follows:chromosome 2;
.. ... ... ...
* From the 1st generation to the 15th generation, thexx1n;Cxx2n;Cyxn;Czxn;/xn;ayxn;azxn;Cyy1n;Cyy2n;Cxyn; mutation range was between3760 and 3760 lm;
* from the 16th generation to the 30th generation, thezyn;/yn;axyn;azyn;Czz1n;Czz2n;Cxzn;Cyzn;/zn;axzn;ayzn:
mutation range was between1880 and 1880 lm;chromosome n:
* from the 31st generation to the 50th generation, the
e15T
mutation range was between188 and 188 lm;