4。1。 SCARA Manipulator

A SCARA planar manipulator with two revolute joints along with the modeling DH parameters are shown in Fig。 6。 The task specification and constraints are given  in Table III。 The constraint values for the design variables, the link lengths and cross- sectional parameters for each cross-section, were defined to be the same for all three optimization approaches。

The results of the optimization process utilizing the three evolutionary techniques are presented in Tables IV—VII。 In these tables, the values of the design variables for each cross-section as well as the torque and number of function evaluations for each technique are presented。 The length values obtained satisfy the kinematic requirements (the manipu- lator reaches the desired points without violating any constraints) and along with the link parameters (cross-section and Young's modulus) the structural requirements。 It is observed that the GAE and DE results are closer to each other as compared to those from the SGA approach。 However, the DE approach consistently obtains smaller tor- ques and smaller number of function evaluations。 The number  of function  evaluations is an indication of the computing effort required in reaching the optimum torque value for the same number of generations。

FIGURE  6    SCARA manipulator  schematic  and  kinematic parameters。

TABLE III    SCARA robot  task specifications  and constraints

Task Constraints

Initial final

position (or) position (or)

Motion  time

Payload Max。  deflection

Joint range (Degrees)

TABLE  IV     Circular  cross-section  (SI units)

DI D2 d1

Torque function

d2 *10E3 evaluations

TABLE  V     Square  cross-section  (SI units)

DI D2 d1

Torque function

d2 *10E3 evaluations

TABLE  VI     C-channel  cross-section  (SI units)

Torque function

LI L2 HI H2 BI B2 TI T2 *10E3 evaluations

TABLE  VII     Rectangular  cross-section  (SI units)

Torque function

LI L2 BI B2 DI D2 b1 b2 d1 d2 *10E3 evaluations

TABLE VIO    SCARA summary results (SI  units)

Circular cross-section

Square cross-section

C-channel cross-section