An integrated process planning and robust fixture design system 6

Other than making the localization errors as small as possible, to make the selected locating features less sensitive to the external errors such as locating surface errors, set-up error

Robust Fixture Layout

6.1 Introduction

A fixture is a device to locate and secure a workpiece to maintain its orientation and position during a machining process using supports, locators and clamping

mechanisms (Nee et al, 1995) The datum reference frame of the workpiece is

established with respect to the reference frame of a machine tool A machined feature may have geometric errors in terms of its form and position in relation to the workpiece datum reference frame A misalignment error between the workpiece datum reference frame and machine tool reference frame is a major cause of the geometric error of a machined feature This misalignment error is known as the localization error which is essentially caused by a deviation in the position of the contact point between a locator and the workpiece surface from its nominal specification It is also highly dependent on the configuration of the locator in terms

of their positions relative to the workpiece A proper design of the locator configuration (or locator layout) would have a significant impact on reducing the localization error This is often referred to as fixture layout optimization (Wang and Pelinescu, 2001) Many research studies have been conducted in searching for feasible or optimal solutions of fixture layout and/or configuration using techniques

such as expert systems (Kumar et al, 1992), case-based reasoning (Kumar et al, 1995),

genetic algorithms (Wu and Chan, 1996), nonlinear-programming (Asada and By, 1985), etc

Other than making the localization errors as small as possible, to make the selected locating features less sensitive to the external errors such as locating surface errors, set-up errors and fixture errors which result in the localization errors is also vital and desirable The issue addressing the robustness of fixture layout has not been studied intensively, but it must be considered because it can differentiate the best solution from several sets of feasible fixture layouts To ensure the feasibility of the location layout, the location points on a locating face should be spaced out as far as possible to ensure location stability and reliability

In this chapter, fixture layout optimization is considered as a three-objective optimization problem The first objective is to maximize the distance between the locating points on a locating face The second one is to minimize the mean localization errors on the machining features in a set-up, and the third one is to minimize the variation of the mean localization errors The non-dominated multi-objective optimization method is applied to achieve the above goals ACO and GA are employed, and the simulation results are compared Through the two approaches, Pareto sets of robust fixture layouts can be obtained

6.2 Fixture Model

6.2.1 Location Layout

According to the 3-2-1 location plan, three location faces with six locators are applied

to locate a workpiece during the machining process For example, as shown in Figure 6.1, a hole and a slot have to be machined from the top face The bottom face, front face and right face are chosen as the locating faces The task for location layout is to select the proper location points on each locating faces (Figure 6.1 (a)) An illustrative example of location layout is shown in Figure 6.1 (b)

(a) Candidate points (b) Selected location points

Figure 6.1 Location layout

Different methods have been addressed for location layout design, and they have been reviewed in Chapter 2 In this study, analysis of the distance of location points and the localization errors are conducted when choosing those points, ensuring that the location points selected are optimal with respect to tolerance analysis and robustness

6.2.2 Sources of Localization Errors

Three sources of errors, i.e., fixture set-up error, locator profile error and datum

profile error, which result in localization errors, are considered in this study They are described in detail in Figure 6.2 As shown in Figure 6.2a, if the localization source error exists due to the setup errors of the locators, the position of the contact point can vary As shown in Figure 6.2b, if another localization source error associated with the default manufacturing error of the locator appears, then the positions of contact points and workpiece will vary The third localization source error, as shown in Figure 6.2c,

is concerned with the size/shape error of the raw workpiece Its presence will not only change the position of contact points with respect to the location point, but also lead

to the translation and rotation of the workpiece

(a) Locator position error (b) Locator profile error (c) Datum profile error

Figure 6.2 Sources of localization error

6.2.3 Mathematical Model of Localization Error

In process planning, a set-up corresponds to a unique fixture layout Since there are usually more than one machining feature to be machined in a set-up, the localization errors will affect all the machining features in this set-up Therefore, to analyze the

Workpiece

Locator

Workpiece Workpiece

Ideal Position Real Position

localization errors for a fixture layout, it is desirable to consider the perturbation of all the machining features in a set-up

To model the localization errors, some assumptions have been made: (1) the workpiece is prismatic and rigid, and the elastic deformation of the workpiece is negligible; (2) the fixture-workpiece contacts are modelled as points without friction; (3) the fixture layout uses the 3-2-1 principle; (4) there is no machine tool error, only the three types of error sources mentioned in Section 6.2.2 are considered

Theoretically, a workpiece should be maintained in a specified position after it has been located in a fixture However, position variation of the locating points with respect to tool-setting points induced by the localization source errors will lead to

deviation of workpiece position The deviation of a workpiece can be defined as δq p = [ = [δx p , δy p , δz p , δα p , δβ p , δγ p]T, inclusive of the part positional T

p

d and rotational error T

where J is the Jacobean matrix and N=diag(n 1 ,…, n 6 ), and n i is the surface normal at

the ith contact point

For a given key point t 3 on the machining feature to be machined in the current

as:

(6.2)

where I 3×3 is the identity matrix, and the ˆ

f

d is a skew-symmetric matrix

In some manufacturing applications, directional deviation is considered, which refers

to the deviation of the point t 3 in a given direction s 3 Thus, the directional point-wise manufacturing error can be obtained from:

(6.3)

For a set of key points P={t i , i=1,…,m} on the machining features in the current setup,

a set of deviation vector S={s i , i=1,…,m} is accompanied with the points The locator

layout for the current setup can be evaluated as:

6.3 Multi-objective Optimization

6.3.1 Formulation of the Multi-objective Optimization Problem

In this approach, the contact point deviations δrs are the noise factors that affect the

fixture design performance It is solved using the Monte-Carlo Simulation (MCS) sampling method In the MCS method, a batch of workpieces is simulated to be located on the designed fixture for a manufacturing process The contact point

deviations are independently generated with Gaussian random distribution N (0, σ2),

where σ is the standard deviation that can be calculated using σ=t/3, where t is the

tolerance for each of them

Based on the MCS method, two objectives, i.e., the mean localization error and the

variation of the localization error, together with the distance of locating points, are formulated according to equation (6.4) They are:

Minimize:

∑

(6.5) ∑ (6.6)

Maximize:

∑ , , , , (6.7)

where, S n is the number of simulation run, F n is the locating face, and i, j are the locating points on F n

6.3.2 Optimization Methods

Multi-objective optimization is a fast growing area of research Being a based approach, GA introduced in Chapter 1 is well suited to solve multi-objective optimization problems A generic single-objective GA can be modified to find a set of multiple non-dominated solutions in a single run The crossover operator of GA may exploit structures of good solutions with respect to different objectives to create new non-dominated solutions in unexplored parts of the Pareto front In addition, most multi-objective GA does not require the user to prioritize, scale, or weigh objectives Therefore, GA has been one of the most popular heuristic approaches to multi-objective design and optimization problems

population-The first multi-objective GA, called vector evaluated GA (or VEGA), was proposed

by Schaffer (1985) Afterwards, several multi-objective evolutionary algorithms were developed including Multi-objective Genetic Algorithm (MOGA) (Fonseca and

Fleming, 1993), Niched Pareto Genetic Algorithm (NPGA) (Horm et al, 1994),

Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele, 1999), improved SPEA (SPEA2) (Zitzler et al, 2001), Pareto-Archived Evolution Strategy (PAES) (Knowles and Corne, 2000), Pareto Envelope-based Selection Algorithm (PESA) (Corne et al,

2000), Region-based Selection in Evolutionary Multiobjective Optimization (PESA-II)

(Corne et al, 2001), Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) (Deb

et al, 2002), Multi-objective Evolutionary Algorithm (MEA) (Sarker et al, 2002),

and Yen, 2003), and Dynamic Multi-objective Evolutionary Algorithm (DMOEA) (Yen and Lu, 2003) Several survey papers (Fonseca and Fleming, 1993; Coello, 2000;Jensen, 2003) have been published on evolutionary multi-objective optimization

In this research, the NSGA-II Pareto ranking algorithm is adopted to solve this objective fixture layout problem

multi-6.3.2.2 Multi-objective Ant Colony Optimization

Given the suitability of stochastic, population-based algorithms to problem domains such as multi-objective optimization (Deb 2002), ACO (Dorigo and Stiitzle 2004) algorithms have been shown to be effective problem solving strategies for multi-objective optimization problem domains, and many existing approaches have been

reviewed by Angus and Woodward (2009)

A diversity of ACO approaches adopted to solve the multiple objective problems The existing algorithms are CPACO (Angus, 2007), MACS (Barán and Schaerer, 2003)

MOAQ (Romero and Manzanares, 1999), MOACOM (Gravel et al, 2002), ACOAMO (McMullen, 2001), SACO (T’kindt et al, 2002), MACS-VRPTW (Gambardella et al, 1999), COMPETants (Doerner et al, 2003) and PACO-MO

(Doerner et al, 2004), MONACO (Cardoso et al, 2003) and ACO-bQAP Ibáñez et al, 2004) A detail review and analysis of existing research on multi-

(López-objective ACO can be found in Garcìa-Martínez et al (2007) and Angus and Woodward (2009)

A multi-objective optimization problem contains several objectives that require optimization In the case of single objective optimization problems, the best single design solution is the goal However, for multi-objective problems with several and possibly conflicting objectives, there is usually no single optimal solution Therefore, the decision maker is required to select a solution from a finite set by making compromises A suitable solution should provide acceptable performance over all the objectives

There are two approaches to solve the multi-objective optimization problem One approach is the classical weighted-sum approach where the objective function is formulated as a weighted sum of the objectives The problem, however, lies in the correct selection of the weights or utility functions to characterize the decision-maker’s preferences In order to solve this problem, the second approach called the Pareto-optimal solution can be applied Pareto-optimal solutions are also called non-dominated solutions Non-dominated ranking is used to evaluate the objectives in this study

Once the solutions have been evaluated, they are sorted using non-domination into each front The first front is the completely non-dominated set in the current population The second front is dominated by the individuals in the first front only and the front continues in this way Each individual in each front is assigned rank

(fitness) values or based on the front which they belong to Individuals in the first front are given a fitness value 1 and individuals in the second front are assigned fitness value as 2, etc In addition to the fitness value, a crowding distance is calculated for each individual The crowding distance is a measure of how close an individual is to its neighbours Large average crowding distance will result in better diversity in the population The ranking process follows three steps:

1 Obtain the 1st rank individuals:

For each individual p in the main population, P does the following:

by p

– for each individual q in P:

* if p dominates q then

· add q to the set Sp

* else if q dominates p then

This is carried out for all the individuals in the main population P

2 Obtain the subsequent rank individuals

• Initialize the front counter to one, i.e., i = 1

– Initialize the set Q for storing the individuals for (i+1) th front

* for each individual q in Sp (Sp is the set of individuals dominated by p):

– Increment the front counter by one

3 Calculate the crowding distance

The crowding distance is calculated as below

– for each objective function m:

* for k = 2 to n

6.3.3 Solving the Multi-objective Optimization Problem

One idea of the study is to generate only feasible solutions To ensure feasible solutions, some fixturing rules have to be applied To achieve this, before performing the optimization, the three candidate locating surfaces are ranked by decreasing surface areas The first one is assigned as the first locating face, and the second and third ones are assigned as the second and third locating faces If the surfaces have the same areas, the sequence is chosen randomly, and accordingly, distance optimization will be performed at the three faces separately For the first face, three locating points are to be searched, and two locating points are to be found at the second and one point

to be searched from the third face The distance of locating points is the sum of the distance among the three points in the first locating face and the distance of the two points in the second locating face

The design space used in both of the approaches is to represent the information for a

set-up, i.e., it contains information of the target machining features and the fixturing

features Each target machining feature is represented by several key points, and the fixturing features are identified by face IDs together with the discretized points The aim is to search for the optimal location points on the fixturing features to achieve a feasible and robust fixture layout with minimal localization errors

A solution contains the decision variables (location information) and the evaluation information The location information is defined as information with six locating points together with the locating faces which the location points belong to The