GA is recognized to be different than traditional gradient based optimization techniques in the following four major ways [10]:

1。 GAs work with a coding of the design variables and parameters in the problem, rather than with the actual parameters themselves。

2。 GAs makes use of population-type search。 Many different design points are evaluated during each iteration instead of sequentially moving from one point to the next。

3。 GAs needs only a fitness or objective function value。 No derivatives or gradients are necessary。

4。 GAs use probabilistic transition rules to find new design points for exploration rather than using deterministic rules based on gradient information to find these new points。

4。 Approach

4。1。 Fixture positioning principles

In machining process, fixtures are used to keep workpieces in a desirable position for operations。 The most important criteria for fixturing are workpiece position accuracy and workpiece deformation。 A good fixture design minimizes workpiece geometric and machining accuracy errors。 Another fixturing requirement is that the fixture must limit deformation of the workpiece。 It is important to consider the cutting forces as well as the clamping forces。 Without adequate fixture support, machining operations do not conform to designed tolerances。 Finite element analysis is a powerful tool in the resolution of some of these problems [22]。

Common locating method for prismatic parts is 3-2-1 method。 This method provides the maximum rigidity with the minimum number of fixture elements。 A workpiece in 3D may be positively located by means of six points positioned so that they restrict nine degrees of freedom of the workpiece。 The other three degrees of freedom are removed by clamp elements。 An example layout for 2D workpiece based 3-2-1 locating principle is shown in Fig。 4。

Fig。 4。 3-2-1 locating layout for 2D prismatic workpiece

The number of locating faces must not exceed two so as to avoid a redundant location。 Based on the 3-2-1 fixturing principle there are two locating planes for accurate location containing two and one locators。 Therefore, there are maximum of two side clampings against each locating plane。 Clamping forces are always directed towards the locators in order to force the workpiece to contact all locators。 The clamping point should be positioned opposite the positioning points to prevent the workpiece from being distorted by the clamping force。

Since the machining forces travel along the machining area, it is necessary to ensure that the reaction forces at locators are positive for all the time。 Any negative reaction force indicates that the workpiece is free from fixture elements。 In other words, loss of contact or the separation between the workpiece and fixture element might happen when the reaction force is negative。 Positive reaction forces at the locators ensure that the workpiece maintains contact with all the locators from the beginning of the cut to the end。 The clamping forces should be just sufficient to constrain and locate the workpiece without causing distortion or damage to the workpiece。 Clamping force optimization is not considered in this paper。 

4。2。 Genetic algorithm based fixture layout optimization approach

In real design problems, the number of design parameters can be very large and their influence on the objective function can be very complicated。 The objective function must be smooth and a procedure is needed to compute gradients。 Genetic algorithms strongly differ in conception from other search methods, including traditional optimization methods and other stochastic methods [23]。 By applying GAs to fixture layout optimization, an optimal or group of sub-optimal solutions can be obtained。

In this study, optimum locator and clamp positions are determined using genetic algorithms。 They are ideally suited for the fixture layout optimization problem since no direct analytical relationship exists between the machining error and the fixture layout。 Since the GA deals with only the design variables and objective function value for a particular fixture layout, no gradient or auxiliary information is needed [19]。