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inverse of the ratio of output to input torques. For any given angular velocity of the input link, this ratio is also a representation of
the angular velocity of the output link, meaning that its minimization could result in lower accelerations and hence lower inertial
forces exerted on the mechanism. It could also be argued that minimizing the ratio of output to input angular velocities would
result in smaller actuating forces. It is then true to say that including the third objective would allow themechanismto be run by a
smaller motor/actuator and to experience smaller loads. According to Freudenstein's theorem [39], the maximum value of this
objective occurs when the collineation axis, the line connecting instant centers P13 and P24, lies perpendicular to the coupler link.
Since P13 and P24 are at the intersections of the link pairs (input, output) and (ground, coupler) respectively, the instant of this
extreme can be easilymonitored and its value computed. Its value is also computed fromthe angular velocity ratio theoremusing
instant centers P12,P24, and P14. So the third objective MAVR can be stated as follows:Here, the strict inequality is considered to make sure that the open and crossed configurations of the mechanism would not
switch during the motion (no change point in the mechanism), meaning that there would be no need to control the operating
mode of the mechanism and hence the resulting mechanisms would be simpler and more practical.
Also, to make sure that the precision points are tracked in the desired order, it is necessary for the corresponding θ2 (i)'s to
follow this order as well. This is equivalent to the following constraint:In the next section, the optimization method used to solve this problem is discussed.
3. Introducing the Pareto GA with Adaptive Local Search (PGAALS)
There are several ways to treat problems with multiple, possibly conflicting objectives. The “conflict” among the objectives
implies that there is usually no single “best” point in the solution space that surpasses all other pointswith respect to all objectives.
Points that surpass others according to some criteria may well fall inferior to them according to other criteria. This brings up the
notion of Pareto-optimality, where multiple, non-dominated solutions, collectively referred to as the “Pareto set” are sought,
rather than a single best point [40]. Pareto-optimality has found wide applications in real world problems, as it provides the user
with the opportunity to choose from a variety of plausible solutions according to his/her preferences.
Many search/optimization algorithms have adopted the notion of Pareto-optimality in their quest for optimal solutions when
multiple measures of desirability need to be considered. With the emergence of Evolutionary Algorithms, EAs, more researchers
were encouraged to employ the notion. Thanks to the population-based structure of EAs which made it possible for multiple
candidate solutions to be processed simultaneously and to their exploration powerwhich considerably reduced the computational
cost of finding non-dominated points.
Being among the first EAs applied to problems withmultiple objectives,MOGAs soonmade their way to the top of users' lists in
practical applications, as their easily tunable genetic operators proved to be quite efficient in spotting the often scattered non-
dominated points. A rank-based fitness assignment scheme was commonly used to map the multi-dimensional space of the
objectives to the uni-dimensional space of fitnesses, where inpidual solutions were “ranked” based either on the number of
solutions they dominated or on the order of the non-dominated layer of solutions they belonged to [31].
Of the many variants of MOGAs, a modified version of Srinivas and Deb's Non-dominated Sorting Genetic Algorithm, NSGA,