Torque

*10E3 function evaluations Torque

*10E3 function evaluations Torque

*10E3 function evaluations Torque

*10E3 function evaluations

SGA 7。90 9777 5。94 9777 5。 50 3。71

GAE 5。38 4801

  4862 3。34 2。 19

DE 3。10 1260

  1200 2。77 1。68

The optimization results summarizing the torque and function evaluations for the SCARA manipulator for the various cross-sections are shown in Table VIII。 The rect- angular cross-section provides minimum torque for the defined task without violating any of the constraints。 The objective function history for the SCARA manipulator for each cross-section is shown in Fig。 7。 It is observed that the DE converges faster than the other two in about  20 generations  and in a monotonic   fashion。

OPTIMUM  ROBOT  DESIGN

FIGURE  7    SCARA manipulator  objective function  history。

4。2。 3-DOF Articulated

A 3-DOF articulated PUMA type manipulator along with  the  modeling  DH  param- eters are shown in Fig。 8。 The last two links were optimized but the objective function included the torque from all  three  joints。  The  task  specification  and  constraints  are given in Table  IX。  The  constraint  values  for  the  design  variables  were  the  same  for all  three  optimization   approaches。

The results of the optimization process utilizing the three evolutionary techniques are presented in Tables X—XII。 In these tables, the values of the design variables for each cross-section and the torque and  number  of  function  evaluations  for  each  technique are presented。 The length values obtained satisfy the kinematic requirements (the manipulator reaches the desired points without violating  any  constraints)  and  along with the link parameters (cross-section and Young's modulus) the structural  require- ments。 It is observed that the DE approach consistently obtains smaller torques  and smaller number of function evaluations。 The number of function evaluations is an indi- cation  of the  computing  effort  required  in  reaching  the  optimum torque。

The optimization results summarizing the torque and function evaluations for the articulated  manipulator  for  the  various  cross-sections  are  shown  in  Table  XIII。 The square cross-section  provides  for the minimum  torque for the defined task   while