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Machine Theory
Mechanism and Machine Theory 43 (2008) 1610–1624
ate/mechmt
Optimum exact/approximate point synthesis
of planar mechanisms
Nadim Diab, Ahmad Smaili *
Department of Mechanical and Mechatronics Engineering, Hariri Canadian University, Meshref, . Box 10 Damour, Chouf 2010, Lebanon
Received 23 May 2007; received in revised form 12 December 2007; accepted 12 December 2007
Available online 15 February 2008
Abstract
An optimization approach is herein presented for mixed exact–approximate dimensional synthesis of planar mecha-
nisms. Any number of points along a closed path may be traversed exactly while the rest approximately. Ant-gradient
search is applied to an objective function based on log10 of the error between the desired positions and the generated ones.
The log10 function results in miniscule errors being generated at the exact points. The algorithm is tested by way of four
case studies in three examples that focus on four-bar and Watt-I six-bar mechanisms involving path and function gener-
ation tasks.
Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Exact/approximate synthesis; Ant-gradient optimization; log10 objective function
1. Introduction
The method applied to a mechanism synthesis problem depends on whether the design points are traversed
precisely or approximately. Thus we speak of exact positions synthesis, approximate positions synthesis and
mixed exact–approximate positions synthesis. Exact position methods [1] generate mechanisms that traverse a
limited number of desired points precisely with no control on the motion between the exact positions. Approx-
imate positions synthesis is required if the number of desired points exceeds a few (five positions for four-bar
motion generator) and requires the use of optimization methods to plish the synthesis task [2–25]. The
drawback in optimum synthesis methods is that the solution will not traverse any of the desired pos