DH-TABLE

FIGURE  8    Articulated  PUMA  type manipulator  schematic and  DH parameters。

TABLE IX    Articulated  manipulator  task specifications  and constraints

TABLE  X     Articulated  circular  cross-section  (SI units)

DI D2 d1

function

d2 Torque evaluations

TABLE  XI     Articulated  square  cross-section  (SI Units)

TABLE XII   Articulated  rectangular  cross-section  (SI units)

function LI L2 BI B2 DI D2 b1 b2 d1 d2 Torque  evaluations

function function function

Torque evaluations Torque evaluations Torque evaluations

p aoo 3oo zdo

'’ 10 15 20 25 30 35 40 46 50

FIGURE  9    Articulated  manipulator  objective function history。

meeting the constraints when compared for all three approaches。 The objective function history for the articulated manipulator for each cross-section is shown in Fig。 9。 It is observed that the DE technique converges faster than the other two in a monotonic fashion and requires  about 20 generations。

4。3。 Additional Discussion of  Results

The results presented  show  the  objective  function  value  (torque  value)  and  number of function  evaluations  for each  evolutionary  approach  for each  cross-section  analyzed。

374 P。S。  SHIAKOLAS  et at。

It is observed that the torque value, the objective function used during the optimization process, for each cross-section is consistent between the three approaches。 The smallest is obtained from DE followed by GAE (GA with elitism) with the largest value given by the simple GA。 This indicates that the traditional GA techniques do not minimize the objective function or design variables even with a larger number of generations。

Another interesting observation is that the number of function evaluations follows a similar trend with the DE reaching a steady state or convergence type of behavior in only 20 generations。 Even though the traditional GA techniques were  allowed  to run for more generations thus increasing the number of function evaluations, their objective functions were not even close to those obtained by    DE。

The large difference in the number of function evaluations required to minimize the objective function is directly related to time savings。 These time savings are amplified considering that the DE operates on real-valued design variables and does not require binary conversion  as the GA。来*自-优=尔,论:文+网www.youerw.com