Engineering Optimization: Theory and Practice, Fourth Edition

536 Geometric Programming

The objective function for minimization is taken as the sum of squares of structural

error at a number of precision or design positions, so that

f=

∑n

i= 1

ε^2 i (E 5 )

wherendenotes the total number of precision points considered. Note that the errorεi

is minimized whenf is minimized (εiwill not be zero, usually).

For simplicity, we assume thata≪dand that the errorεiis zero atθ 0. Thus

ε 0 = at 0 θi=θ 0 , and Eq. (E 3 ) ieldsy

K= 2 cdcosφdi + 2 accosθ 0 cos (φd 0 −θ 0 ) − 2 adcosθ 0 (E 6 )

In view of the assumptiona≪d, we impose the constraint as (for convenience)

3 a

d

≤ 1 (E 7 )

where any larger number can be used in place of 3. Thus the objective function for

minimization can be expressed as

f=

∑n

i= 1

a^2 os(c θi− osc θ 0 )^2 − 2 ac(cosθi− osc θ 0 )( osc φdi− osc φd 0 )

c^2 sin^2 φdi

(E 8 )

Usually, one of the link lengths is taken as unity. By selectingaandcas the design

variables, the normality and orthogonality conditions can be written as

∗ 1 +∗ 2 = 1 (E 9 )

2 ∗ 1 +∗ 2 = 0 (E 10 )

2 ∗ 1 + 0. 5 ∗ 2 +∗ 3 = 0 (E 11 )

These equations yield the solution∗ 1 = − 1 ,∗ 2 = , and 2 ∗ 3 = , and the maximum 1

value of the dual function is given by

v(∗)=

(

c 1

∗ 1

)∗ 1 (

c 2

∗ 2

)∗ 2 (

c 3

∗ 3

)∗ 3

(E 12 )

wherec 1 ,c 2 , andc 3 denote the coefficients of the posynomial terms in Eqs. (E 7 )

and (E 8 ).

For numerical computation, the following data are considered:

Precision point,i 1 2 3 4 5 6

Input,θi(deg) 0 10 20 30 40 45

Desired output,φdi(deg) 30 38 47 58 71 86

If we select the precision point 4 as the point where the structural error is zero (θ 0 = 03 ◦,

φd 0 = 85 ◦), Eq. (E 8 ) ivesg

f= 0. 1563

a^2

c^2

−

0. 76 a

c

(E 13 )