神兜兜和他的朋友们

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38 A. Caballero-Ruiz et al. / Mechatronics 17 (2007) 231–243
a) The MMT goes to its home position. ofB2).Thisevaluation ismadeapplyingthefollowingapti-
b) B1 moves to a position under B2, and then B1 moves tude function:
forward along Y axis until electrical contact with B2
m 2
Xq??????????????????????????????????????????????????????????????????????????????2 2 2
is made. Its position is recorded in a text file. erri 1/4 eXlX0iT teYlY0iT teZlZ0iT eRtrT
l1/41
c) B1 moves back a distance dY along Y axis, which
allows the error generated by backlash to be elimi- l 1/4 1;2;3;...;m; i 1/4 1;2;3;...;n e11T
nated. It moves then up a distance dZ.
where erri is the quadratic error that represents the adapt-
d) B1 moves forward again along Y axis until electrical
ability of each chromosome to the sphere defined by the
contact with B1 is made and the program records its
experimental data; Xl, Yl, Zl represent the coordinates of
position.
the contact points where B1 touched B2; m is the number
e) The steps from b to d are repeated along the Z axis
of experimental data; R is the radius of B1 and r is the ra-
(Fig. 10).
dius of B2.
In each generation, the aptitude function is calculated
Fig. 11 shows the setup for the evaluation method and a
for each chromosome and the k best chromosomes are
plot of the data obtained after a measurement process.
selected to produce the next generation of chromosomes.
In this case, we selected the mutation as genetic operator;
5.1. Determination of the center of B2 and its position errors
in other words, to produce the entities of a new generation,
the fathers are a?ected by a randomly mutation factor.
The obtained measurements for each ball position are
Each selected chromosome is considered as a father of
processed with a special algorithm to determine the ball
some of the next generation entities, where each father cre-
center and the position errors. There are numerical meth-
ates s =n/k sons.
ods to obtain the center of a sphere associated to several
points of its surface, i.e. least square method. In this X0eepkTthT 1/4 X0h trandheDXT;
research, we use a GA, which is a method of adaptive Y0eepkTthT 1/4 Y0h trandheDYT; p 1/4 1;2;...;es1T; h 1/4 1;2;3;...;k
search and optimization. The GA mimics some rules of Z0eepkTthT 1/4 Z0h trandheDZT;
the natural evolution to find the optimal solution of a
e12T
problem [30]. From a population of parameter vectors that
are obtained at random, the algorithm simulates the natu- where X0((p*k)+h), Y0((p*k)+h), Z0((p*k)+h) represent the
ral selection to find the best fit for an aptitude function. descendants of the previous generation, DX, DY, DZ are
In order to determine the errors of the MMT related to the mutation factor, and rand(u) is a random number in
the center of B2, a population of chromosomes is created, the range [0,u].
such a population is composed by a triad of values X0, Y0, ThepointsX0((p*k)+h),Y0((p*k)+h),Z0((p*k)+h) fromtheEq.
Z0 (Eq. (10)). Each chromosome represents the possible (12) and the k fathers of the current generation will con-
center of B2. form the next generation of chromosomes X0i, Y0i, Z0i.
The algorithm searches during a defined finite number of
X01;Y01;Z01chromosome 1
generation such values of X0i, Y0i, Z0i where the error is
X02;Y02;Z02chromosome 2 minimal. After a defined number of generations the chro-
e10T
... ... ... ... mosome, where Eq. (11) is minimal, is considered as the
center of B2 eX ;Y ;Z T.
0 0 0
X0n;Y0n;Z0nchromosome n;
After the first algorithm determines the center of B2, the
wheren represents the numberof chromosomes ofthe pop- equations that represent the model of deviations existing in
ulation. The population of the GA is randomly selected in the MMT are introduced to compensate the experimental
the beginning of the algorithm. The next step is to evaluate measurements. The error equations used in this algorithm
which chromosomes are near to the target point (the center are presented below:
Fig. 11. (a) Balls placement in the MMT workspace and (b) plot of the measurements.