The analysis routines related to manipulator motion analysis are inverse kinematics, joint space trajectory generation, and dynamics。 These are evaluated using standard robotic analysis algorithms [21]。

• Inverse kinematics evaluates the required joint variables 8,erai and 8r。or based on the desired  Cartesian coordinates  of the initial and final  positions。

Design Variables From GA/DE

Inverse Pict: & Plzce position

FIGURE   2     Analysis  flow chart。

Joint space trajectory generates the position 8(t) (cubic  polynomial  profile), velocity 8'(/) and acceleration 8”(t) profiles based on the desired motion time  and  joint variables。

Dynamics evaluates the inpidual joint toi ques based on the structural chin acteris- tics of the links, the payload and the position, velocity and acceleration profiles。 The evaluation of the dynamic equations  could  include  also friction  losses  proportional to the magnitude of the velocity。 In this work, the friction losses are not considered in the analysis since the friction coefficients are not known  and their  influence will  be to increase the value of the too que requiied for the motion。 The dynamics  one eval- uated  according  to  the Newton—Euler  method [21]。

                                         (6)

Finite  Element  Analysis  for Deflection

The deflection analysis evalti:ites the deflection of the end effector of the manipulator considering the structural cli:u acteristics of the links, the mateiial properties foi inertia evaluation and the payload。 This calculation  is  performed  using  the  finite  element method [22]。 The maximum deflection occurs when the manipulator is at its maximum reach。  The  generalized  load vector  is derived  using  the paylodd   Fp     d the properties of the links of the manipulator such as cross-section, modulus  of  elasticity,  E,  and density, p。

The links of the manipulator are modeled as beams with the general 3-D beam element  stiffness matrix  [22]。 The structural model  of a two link planar   manipulator is shown in Fig。 3 where the degrees of freedom for each node  are the moments,  M, due to bending about the c-axis and forces    due to axial loading along the y-axis。

The assembled  global  load  vector,  P, for the  two-element  manipulator  after reduction

due to motion  constraints  at  degrees  of  freedom  Mb  and f    is  given by

where J, is the link weight  per  unit  length given by f ——  (p,N,L,g)/ L,

FIGURE   3     Forces  and  moments  for  structure l analysis。

The stiffness matrices for each element are assembled to generate the global stiffness matrix。 The assembled global stiffness matrix for the two-element manipulator, K, after reduction  due  to  constraint  degrees  of freedom  is  given by

The  deflection  vector  v  = {y  , 8  , y  , 8 } is  then  evaluated  according to

U —— K —'P (9)

3。 OPTIMIZATION APPROACHES

Three evolutionary based optimization approaches were investigated, a SGA, simple GAE and DE。