OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
2. Theoretical analysis
2.1. Vector loop analysis of 4R linkage
Fig.1 shows the 4-bar linkage with vector loop. There are 10 independent variables, L1 , L2,
L3
, L4,
L5
, β , θ
2,
θ1,
L0,
L6
.
L
Where 5
and β L5
are two parameters to locate the coupler point. the distance from a convenient reference
point, and the angle that the line AP makes with the line of centre of the coupler AB.
The loop closure equation is:
L + L + L + L 0
(1)
1 2 3 4 =
Using Euler equivalence Eq. (1) becomes:
Fig.1. 4-bar Linkage: Vector Loop and Variables
iθ1
iθ
2
iθ
3
iθ
4
L e + L e + L e + L e 0
(2)
1 2
3
4 =
The coordinates of coupler C, in the reference frame OXY, are,
P X = L 2 cos θ 2 + AP cos ( θ 3 + β )
(3)
P Y
= L
2
sin θ
2
+ AP sin ( θ
3
+ β )
(4)
Eqs. (3) and (4) are used in Eqs. (A1) and (A2) to develop the first part of the goal function.
Fig.1.shows the variables of the 4R linkage.
2.2. Optimization Techniques
When the objective function is nonlinear and non-differentiable, direct search approaches are the methods of
choice. The best known of these are the algorithm by Nelder and Mead, genetic algorithms, by Hook and Jeeves,
evolutionary algorithms. Here we have used evolutionary algorithm.
Users generally require that a practical optimization technique should fulfill three requirements. First, the
method should find the global minimum, regardless of the initial system parameter values. Second, convergence
should be fast. Third, the program should have a minimum number of control parameters so that it will be easy
to use. The optimization method presented here i.e. the DE possesses all the above mentioned properties.
The main steps of DE algorithm are: Initialization, Evaluation, Repeat, Mutation, Recombination, Evaluation,
Selection, (Until termination criteria are met).
In DE an optimization task consisting of ‘D’ parameters can be represented by a ‘D’ dimensional vector; a
population of ‘NP’ solution vectors is randomly created at the start; this population is successfully improved by
applying mutation, crossover and selection operators.
2.3. The Objective Function
The objective function is the sum of two terms. The first part computes the position error (also called the
structural error) as the sum of squares of the Euclidian distances between each desired points along the coupler
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