The constraints for this optimization  include  the deflection  of the end effector  of the manipulator, physical constraints such as the limits on the joint values, and the structural characteristics  of each link。

The deflection 5 is evaluated using finite element analysis techniques when the manipulator is at its maximum reach (completely stretched  out) since this  will  yield the maximum deflection。 Any other configuration will yield a smaller deflection value considering that the same payload is carried。 The deflection evaluation is a function of the structural  and material  properties  of the links  and the payload。

The constraints on the joint limits or range of motion  of the manipulator  actuators are imposed due to physical  constraints。 The joint  constraints  are defined  as

where 8, is the joint value for joint i and 8 c fi" with n being the number of joints。 This constraint is important in selecting a unique solution in the cases where the inverse kinematics solution process (a function of the Cartesian position and link length) yields multiple solutions。

The structural characteristics of the links are  also  included  in  the  constraints。 These structural characteristics are the link lengths and the cross-sectional area charac- teristics。 These constraints are defined  as

where d indicates they structural characteristic, and d e P“with in being the number of structural characteristics。 The design space for the structural characteristics always consists of the link length with the remaining parameters depended on the type of cross-section being considered。 In this work, the cross-sections considered are hollow rectangular, square, cylindrical and  C-channel  and  are shown  in Fig。  1。 The number of the structural constraints is dynamic and depends  on  the  type  of  cross-section being considered in the analysis。 In addition, physical structural  constraints  are imposed during the analysis in the sense that the inner cross-sectional dimensions cannot  be equal or larger  than the  outer dimensions  shown in  Fig。  1。 For    example,

Hollow Rectangular Hollow Circular

C-channel Hollow Square

FIGURE   1     Cross-sections  considered  in  the analysis。

if a manipulator is to be designed with a link having a C-channel cross-section, then the additional dynamic constraints will include t < h and t < b/2。

• Analysis  Procedure

The first step in the analysis process is to define the problem, define the design vari- ables, assign values to all the parameters and define the constraint vector。 The problem definition consists of defining the kinematic structure of the manipulator to be analyzed using Denavit Hartenberg (DH) parameters, the desired initial and final position in Cartesian space and desired time for the motion, the payload, and the material property and cross-section  type of the links。 The schematic of this process is shown in Fig。   2。

The defined values are used in the analysis routines to obtain values for the design variables。 The design variables are checked for constraint violation and then used in evaluating the objective function。 These evaluations are used in the optimization routine where new values for the design variables are generated。 One function evaluation is com- pleted when one set of design variables is analyzed。 The calculated joint torque provides a fitness evaluation for optimization。 The fitness evaluation process is presented later。 This process is repeated until certain criteria are met such as the number of  generations。