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Four-bar linkage
The four-bar mechanism is a very versatile linkage, which has been used extensively in a variety of machines, e.g. the Ackerman steering mechanism in automobiles, sewing machines, earth movers, packaging machines, automobile suspensions etc.
Points on the coupler of a four-bar mechanism follow fascinating paths of order six (6 degree curves). This property is exploited to materialize designs pertaining to path generation. Given any arbitrary closed path, the objective then is to come up with a four-bar mechanism that can generate this path as a coupler curve. As indicated earlier, it is not an easy task to design a mechanism with any desired coupler curve analytically. Hence, we present a computational approach to solve this problem.
We shall deal with a special class of four-bar mechanisms which satisfy the Grashoff's criterion. Crank-rocker type and crank-crank type mechanisms that fall in this category shall be used for the genertaion of closed coupler curves.
A four-bar linkage mechanism is characterized by the following five parameters: L2, L3, L4, r and f. All lengths are normalized with respected to the fixed link. This set of parameter completely characterizes the four-bar and can be used to represent it.
Genetic Algorithms One such set of parameters that uniquely describe a mechanism is called a PHENO_TYPE. Corresponding to each PHENO_TYPE there exists a GENE or GENO_TYPE, which contains all the information of the PHENO_TYPE in a binary-coded form. This coded string of binary bits is the GENE of the mechanism.
A GA program randomly generates an initial population (100 in our case) of Genes. Each of these is decoded to the Phenotype. For each Phenotype the program computes the coupler curve. Now, this coupler curve is matched with the input required curve (using the pattern matching algorithm explained below) and is accordingly allotted a fitness value. Based on these fitness values, a random group of 50 genes is selected from the population. This is called the tournament selection. Of these 50, the best fitting genes are made the parents, they are mutated and crossed in a controlled fashion, thereby generating a new population. Subsequently, this process is repeated many times. After each generation the best genes are retained. The Law of Natural Selection and Survival of the Fittest work and finally a GENE with a very high fitness value evolves. This gene is decoded to give the corresponding Phenotype, which is the final SOLUTION.
Pattern Matching To determine the fitness of a gene, a pattern-matching algorithm is employed. ‘Any closed curve is uniquely represented by it curvature signature, which is a record of the variation of the local curvature with the arc length.’ This is a very useful piece of information which we exploit for the purpose of pattern matching.
Starting from the left most point on the curve and moving along the arc length, we record the local slope. The variation of theta with arc-length s, at a given point on the curve, is the local curvature. The curvature, dq /ds, is plotted vs. s to obtain a curvature plot. We note the following properties about the curvature plot,
The most critical property of this plot is that it uniquely represents the shape of any given curve and hence is termed as the ‘signature plot’. All these properties make this particular signature plot an ideal means for pattern matching. If the signature plots corresponding to two curves match, this straightaway implies that the two physical curves match. Using this pattern-matching strategy, any combination of the following configurations will give a successful match
Two similar curves with different sizes Two similar curves that are translated in space Two similar curves that are relatively rotated in the space
Conclusions and Results
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