3。2。 Differential Evolution

The DE approach contains the same processes of population initialization, mutation, crossover  and  selection。  It  emphasizes  direct  use  of  the  objective  function。  A cost

TABLE I GA  control parameters

Crossover probability 0 6。6

Mutation probability 0。06 6。64

Population size 60 45

No。 of generations 50 36

FIGURE  4    General  flow  of  operations using GAs for optimization。

function, C, is used to rate the inpidual vectors according to their  capability  to  mini- mize the objective function, f  。

Solution vectors without constraint violations have cost functions equal to the objec- tive function。 Any constraint violation tends to increase the cost of a vector by a value I greater than the objective function。 For our study of torque minimization, the lower the cost, the better  or more fit the design variables   are。

The genetic operation of mutation in the DE uses the vector differentiation method (adding the weighted difference between two population vectors to the third vector) to generate a new vector。 DE is a parallel search method that operates on D-dimensional parameter vectors, where D is the number of design variables。 The number of vectors is equal to user defined population size。 The initial vector population is chosen randomly。 The DE process starts from selecting a target vector。 Then, it randomly selects  two other vectors and generates a difference vector, which is multiplied with a user  defined论文网

weighting factor F to obtain “weighted difference vector”。 The weighted difference vector and randomly chosen mutation vector create a noisy vector, which is subjected to crossover process with the target vector in order to generate the trial vector [17,25]。 The trial vector is then compared with the original target vector。 The vector having less cost function is allowed into the new population。 A schematic of the DE process is shown in Fig。 S。

One of the advantages of DE is that it uses real variables as compared to binary numbers like the GA。 This scheme encodes variables as floating point numbers without ever being reformatted as chromosomes with a different binary representation。 Floating- point numbers use the computer resources efficiently [18]。 In our work, we used the con- trol parameters suggested by Storn [26] and shown in Table II。 The population size is generally varied according to the number  of design  variables。

FIGURE   5     Schematic  of  Differential  Evolution  process [25]。

TABLE II   Differential  Evolution  control parameters [26]

Crossover probability 0 5

Weighing factor 08

Population size 60

No。  of generations 20

4。 APPLICATION  TO  MANIPULATOR  DESIGN

The evolutionary techniques introduced were used in the design of SCARA and articu- lated manipulators。 In this section, the results of applying these techniques for manipu- lator design will be presented and discussed。 The manipulator models, kinematics, inverse kinematics and dynamic analyses for both configurations are based on the modified DH parameters [21]。 In the presented examples, the parameters are expressed in the SI system and the angles in  degrees。