Evolutionary based techniques  are  employed  in  order  to  overcome shortcomings of  conventional  optimization  methods  [2,8—10]。  GA  and  their  variants   are  part of  the  evolutionary  techniques  and  have  been  extensively   used   for   modular robot design [I I—14], inverse and forward  kinematics  [9,15] and  optimum motion and path planning [10, 16]。 DE is  a  recently  developed  evolutionary technique [17,18]。 DE has been successfully applied  to  the  optimum  design  of  digital  filter and communication control [17], and according to information on  the web  [19] to many perse domains such as dynamic systems, controller design, heat  transfer analysis and design, and many more。 The DE technique to the best of our knowledge has never been investigated or applied to the optimum design of serial link manipulators。

This article addresses the application and comparison of three evolutionary techniques for optimum design of serial link robotic manipulators based to task specifi- cations。 The objective function minimizes the required torque for a defined motion subject to various constraints while considering kinematic, dynamic and structural conditions。 The kinematic and dynamic analyses are derived based on robotic concepts and the structural analysis is performed based on the finite element method。 The design variables examined are the link parameters and the link cross-sectional characteristics。 The developed environment was employed in optimizing the design variables for a SCARA and  an articulated  3-DOF PUMA  type  manipulators。

2。 OPTIMIZATION PROBLEM DEFINITION

The general  optimization  problem  is defined  as follows   [20]

where f(:x) is the objective function, g;(x) is the set of inequality constraints and x o  fi"is the real-valued  design  variable  vector  with  n  being  the number  of design  variables。

The optimization problem investigated is that of optimizing the link parameters for a robotic manipulator to obtain minimum power requirements or joint torques。 The task specification is subpided into two elements, the kinematic characteristics (the required position of the end effector) and dynamic characteristics (the time required to complete the motion while carrying the payload and considering the inertia properties of the links themselves)。 Constraints are imposed  on the minimum  and maximum  (range) values of the link parameters that include the link length, and the link cross-sectional area characteristics, and the allowable deflection of the end effector。 In addition, kinematics constraints are imposed that address physical limitations such as the range of motion of the  actuators  of the manipulator。

• Objective Function

The objective function is defined as the cumulative sum of the torques for each joint during  the  motion   of  the manipulator,

time joints

where  /,  is the torque at time  i for joint j。

• Constraints

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。  Other  constraints  that  could  be  included for the design of manipulators are quantitative measures of various workspace attributes such as manipulability  or the structural length  index。