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It has been argued that the multiple objectives of the problem are “conflicting”; meaning that the problem generally has no
single, globalminimum, solution [16,19]. This implies that we should be seeking a set of Pareto-optimal solutions, where no single
solution could be found to surpass all other solutions with respect to all objectives. Instead, one would be dealing with multiple,
non-dominated solutions where any attempt by a Pareto-solution to dominate another solution with respect to a certain objective
would result in the first solution to fall inferior to the second with respect to, at least, another objective [21–28].
The implementation of the Pareto-based optimal design of mechanisms for path generation purposes requires an efficient
Multi-Objective Optimization, MOO, tool. Several such tools have been developed over the past two decades, including Schaffer's
Vector Evaluated Genetic Algorithm, VEGA, [29], Fonseca and Fleming's Multi-Objective Genetic Algorithm, MOGA, [30], Srinivas
and Deb's Non-dominated Sorting Genetic Algorithm, NSGA, [31], Zitzler and Thiele's Strength Pareto Evolutionary Algorithm,
SPEA, [23], Knowles and Corne's Pareto Archived Evolution Strategy, PAES, [24] and Horn et al's Niched–Pareto Genetic Algorithm,
NPGA, [22]. A comprehensive review of these and other MOO methods can be found in [25–28].
This paper presents a new formulation for the problem of linkage design for path generation purposes. The problem is
formulated as a triple-objective optimization problem. These, often conflicting objectives are TE, TA and MAVR. They are
formulated as functions of the design variables (lengths, angles and coordinates). Topological and functional constraints are
presented to help identify the feasible region of the solution space.
An NSGA-based hybrid algorithm called the Pareto Genetic Algorithm with Adaptive Local Search, PGAALS, is also introduced
and its constituting elements are described in some detail. The algorithm combines the exploration power of traditional NSGA
algorithm with the exploitation capability of an adaptive local search that uses an adaptive strategy to set the radius of the local
search neighborhood at each search step.
To demonstrate the applicability and efficiency of the proposed approach and the new search algorithm, the popular problem
of designing a four-barmechanismto generate a straight line is investigated. First the problemis solved separately for one and two
objective functions and results are compared with those reported in the literature. The results show that the proposed search
algorithm is capable of generating distinctly better solutions compared to other MOO algorithms. The problem is then solved for
three objective functions and the quantitative and qualitative significance of the resulting solutions are discussed.
2. Problem formulation
Four-bar linkages owe their popularity to their structural simplicity and functional versatility. Comprising a low number of
components and revolute joints thatmakes themeasy tomanufacture, assemble andmaintain, they can performa variety of tasks,
including motion [32–34] function[35], and path generation [7–19] with reasonably good precision. That is why many designers
prefer them over more complex linkages that perform the same tasks only more accurately.Now if length r2 , angle ν and the product r4.sinγ in Fig. 2 are kept constant, we can still have various values for TA and r4 for the
same MA[38], implying that TA and MA may vary independently of one another, resulting in linkages with good TA and poor MA
and vice versa, despite the fact that both parameters are functions of the linkage configuration. The above argument is also
supported by numerous simulation results partially presented later in this article.
We further assert that instead of MA, one could minimize the ratio of output to input angular velocities which is equal to the