An integrated process planning and robust fixture design system 6

Chapter 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), 116 Chapter Robust Fixture Layout 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 multiobjective 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. 117 Chapter Robust Fixture Layout 6.2 6.2.1 Fixture Model 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. 118 Chapter Robust Fixture Layout 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. Workpiece Locator Workpiece Workpiece Locator Locator Ideal Position (a) Locator position error Real Position (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 119 Chapter Robust Fixture Layout 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 δqp = [ = [δxp, δyp, δzp, δαp, δβp, δγp]T, inclusive of the part positional  dTp and rotational error  θTp . It is mainly caused by contact point error δr = [δr1, …, δr6]T. Under the deterministic localization, the equation can be expressed as (Qin et al, 2006) : (6.1) where J is the Jacobean matrix and N=diag(n1,…, n6), and ni is the surface normal at the ith contact point. For a given key point t on the machining feature to be machined in the current setup, its positional deviation δt caused by the workpiece error δqp can be calculated 120 Chapter Robust Fixture Layout as: (6.2) where I 3×3 is the identity matrix, and the dˆ f is a skew-symmetric matrix. In some manufacturing applications, directional deviation is considered, which refers to the deviation of the point t in a given direction s . Thus, the directional point-wise manufacturing error can be obtained from: (6.3) For a set of key points P={ti, i=1,…,m} on the machining features in the current setup, a set of deviation vector S={si, i=1,…,m} is accompanied with the points. The locator layout for the current setup can be evaluated as: (6.4) 121 Chapter Robust Fixture Layout 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, Sn is the number of simulation run, Fn is the locating face, and i, j are the locating points on Fn. 122 Chapter Robust Fixture Layout 6.3.2 6.3.2.1 Optimization Methods Multi-objective Genetic Algorithm Multi-objective optimization is a fast growing area of research. Being a populationbased approach, GA introduced in Chapter 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 multiobjective design and optimization problems. 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), Weight-based Genetic Algorithm (WBGA) (Hajela and Lin, 1992), Random Weighted Genetic Algorithm (RWGA) (Murata and Ishibuchi , 1995), Non- dominated Sorting Genetic Algorithm (NSGA) (Srinivas and Deb, 1994), Strength 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 123 Chapter Robust Fixture Layout et al, 2002), Multi-objective Evolutionary Algorithm (MEA) (Sarker et al, 2002), Micro-GA (Coello et al, 2002), Rank-Density Based Genetic Algorithm (RDGA) (Lu 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 multiobjective fixture layout problem. 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 multiobjective 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 (Guntsch and Middendorf, 2003), BicriterionAnt (Iredi et al, 2001), ParetoACO (Doerner et al, 2004), MONACO (Cardoso et al, 2003) and ACO-bQAP (LópezIbáñez et al, 2004). A detail review and analysis of existing research on multi- 124 Chapter Robust Fixture Layout objective ACO can be found in Garcìa-Martínez et al (2007) and Angus and Woodward (2009). 6.3.2.3 Non-dominated Sorting 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 decisionmaker’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 nondominated 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 125 Chapter Robust Fixture Layout (fitness) values or based on the front which they belong to. Individuals in the first front are given a fitness value 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: – Initialize Sp which would contain all the individuals that are being dominated by p. – Initialize Np=0. This would be the number of individuals that dominate p. – for each individual q in P: * if p dominates q then · add q to the set Sp. * else if q dominates p then · increment the domination counter for p, i.e., Np = Np + 1. – if Np=0, i.e., no individuals dominate p, then p belongs to the first front; Set rank of individual p to one, i.e., prank = 1. Update the first front set by adding p to the front one F1. 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. • The following steps are carried out while the ith front is non-empty. 126 Chapter Robust Fixture Layout – Initialize the set Q for storing the individuals for (i+1)th front. – for each individual p in front Fi: * for each individual q in Sp (Sp is the set of individuals dominated by p): · if Nq I(k), Ik=I(k+1); else Ik=I(k); if k τ0 because the Pareto set has been improved, the pheromone trail is reinitialized with the new value τ0’ to improve the exploration; otherwise, exploitation is favoured by updating the pheromone trail globally with each solution of the current Pareto optimal set, using the following equation: 1⁄ (6.10) : is the locating distance of the solution in the Pareto set; : is the locating error of the solution in the Pareto set; : is the deviation of the locating errors of the solution in the Pareto set; i: is the selected node in the solution in the Pareto set. 134 Chapter Robust Fixture Layout τ0 is initially calculated, in a similar way of equation (6.9), according to: where (6.11) represents an initial estimation of the locating distance, while represents an initial estimation of the locating error and is the initial estimation of the deviation of the locating error. 6.3.3.3 Performance Metrics To evaluate the experimental results using the two described methods in the preceding sections, the true Pareto Front ( used to calculate ) has to be calculated. A six-step procedure is : 1) Each algorithm is run three times in an iteration. 2) For each algorithm, three sets of non-dominated solutions: , and , are calculated, one for each time. is obtained based on equation (6.12). 3) For each algorithm, overpopulation (6.12) where, i is the number of times. 4) Dominated solutions are deleted from using each algorithm, as 5) A set of solutions and , obtaining the Pareto Front calculated . is obtained as in equation (6.13): 135 Chapter Robust Fixture Layout (6.13) 6) Dominated solutions are deleted from , and an approximated true Pareto Front is finally created. To compare the two algorithms, the overall non-dominated solutions ratio which denotes the ratio between the number of solutions in solutions in and to the number of is calculated. Ideally, a good solution should have a value close to 1. To further evaluate the two algorithms, after each time in an iteration, the results are compared in a similar way, while the Pareto set is obtained according to equation (6.14). Through this comparison, whether an algorithm can provide good solutions continuously but not occasionally can be observed. (6.14) Another factor which affects the performance of an algorithm is whether it can generate good results in a shorter time. Therefore, the time with which the two algorithms are able to generate the Pareto set is also compared after each iteration cycle. 136 Chapter Robust Fixture Layout 6.4 6.4.1 Experimental Results Experiments An illustrative example is presented in this section to demonstrate the proposed approaches and the test results. It is a simplified front knuckle for a generic automotive chassis system. The raw part is a casting followed by machining. Figure 6.6 gives the details of the cast and machined parts. It is assumed that the set-up planning has been completed, and the next stage is to make the location layout for the set-up which is to machine features A and B, having features X, Y and Z as the datum. a) Cast part b) Machined part Figure 6.6 Example part 137 Chapter Robust Fixture Layout The candidate location points on the datum are generated as shown in Figure 6.7. Face X Y Z X Nodes 153 81 57 Z Y Figure 6.7 Candidate points 6.4.2 Discussion To compare the two approaches, five iterations with the iteration number of 40, 100, 200, 500 and 1000 respectively are selected to test the results. For each iteration, three cycles are executed, and the Pareto sets of each cycle for both the algorithms are stored and are compared. The cross probability used in GA is 0.9 and the mutation probability is 0.05. The ant number used in ACO is 20, while the heuristic values are: α is 1, β is 0.3 and ρ is 0.35. For both algorithms, the population of the Pareto set is set to 20. Tables 6.1-5 show the comparisons between the solutions found with the implemented algorithms (GA and ACO) with respect to at different iterations. The results show that the ACO always provides better results than GA, either for the complete 138 Chapter Robust Fixture Layout runs or comparison after each cycle. In addition, ACO can generate good results with less time. Table 6.1 Results of iteration=40, p=20 Total Algorithm N ratio GA 45% ACO 11 55% t(s) 112.5 57.3 1st run N ratio 45% 11 55% 2nd run N ratio 35% 13 65% 3rd run N ratio 40% 12 60% Table 6.2 Results of iteration=100, p=20 Total Algorithm N ratio GA 25% ACO 15 75% t(s) 267.9 143 1st run N ratio 25% 15 75% 2nd run N ratio 35% 13 65% 3rd run N ratio 30% 14 70% Table 6.3 Results of iteration=200, p=20 Total Algorithm N ratio GA 20% ACO 16 80% t(s) 540.8 319.3 1st run N ratio 35% 13 65% 2nd run N ratio 25% 15 75% 3rd run N ratio 30% 14 70% Table 6.4 Results of iteration=500, p=20 Total Algorithm N ratio GA 35% ACO 13 65% 1st run t(s) N ratio 1333.3 40% 835.9 12 60% 2nd run N ratio 45% 11 55% 3rd run N ratio 35% 13 65% Table 6.5 Results of iteration=1000, p=20 Total Algorithm N ratio GA 35% ACO 13 65% 1st run t(s) N ratio 2561.5 35% 1067.8 13 65% 2nd run N ratio 45% 11 55% 3rd run N ratio 45% 11 55% 139 Chapter Robust Fixture Layout An example of the location plan which is selected from the Pareto set is shown in Table 6.6 and Figure 6.8. The values of the location error, the deviation of the location error and the distance of the location points are shown in Table 6.7. Table 6.6 Location layout Fixturing features Location points (-88.108 -143.495 80.000); (43.095 -77.744 80.000); (-57.585 -10.612 80.000) (-93.478 -73.005 52.560); (-93.478 -15.271 45.039) (34.148 2.012 37.279) Location face Face X Face Y Face Z Table 6.7 Objective values Machining Features Face A Face B Orientation (0, 0, -1) (0, 0,-1) Mean location error 1.1E-2 Variation of location error 4.6E-3 Distance of location points 462.3 Figure 6.8 Location layout of example part 140 Chapter Robust Fixture Layout 6.5 Summary Genetic algorithm and ant colony optimization are applied in this study to solve the multiple objective fixture layout problem, i.e., to maximize the distance of the location points, to minimize the localization error and the deviation of the localization error. The experimental results show that both ACO and GA are effective approaches in solving the multi-objective fixture layout problem, and ACO is more attractive by comparing the Pareto set and the computational time. This approach is applicable to the fixture layout problems where the workpiece is prismatic and rigid, the contact between the fixture and workpiece can be taken as point contact without friction, and the machine tool error can be ignored. More effort is needed if it is to be applied to other types of problems. The selection of the locating surfaces in the 3-2-1 method is based on the areas of the surfaces. That may not be applicable to some problems and additional rules may need to be considered. 141 [...]... Figure 6. 4(a) and 6. 4(b) parent1 5 1 5 5 6 6 3 40 65 4 23 55 Parent2 5 5 5 6 6 3 8 32 93 10 60 8 5 1 5 5 6 6 3 40 65 10 60 8 5 5 5 6 8 32 93 4 6 3 23 55 child1 child2 (a) crossover at point level (right side) parent1 5 1 5 5 6 6 3 40 65 4 23 55 Parent2 5 5 5 6 6 3 8 32 93 10 60 8 5 5 5 6 6 8 32 93 4 23 3 child1 55 5 5 5 6 6 3 1 40 65 10 60 8 child2 (b) crossover at point level (left side) Figure 6. 4 Genetic... 13 65 % 1st run t(s) N ratio 2 561 .5 7 35% 1 067 .8 13 65 % 2nd run N ratio 9 45% 11 55% 3rd run N ratio 9 45% 11 55% 139 Chapter 6 Robust Fixture Layout An example of the location plan which is selected from the Pareto set is shown in Table 6. 6 and Figure 6. 8 The values of the location error, the deviation of the location error and the distance of the location points are shown in Table 6. 7 Table 6. 6 Location... set-up planning has been completed, and the next stage is to make the location layout for the set-up which is to machine features A and B, having features X, Y and Z as the datum a) Cast part b) Machined part Figure 6. 6 Example part 137 Chapter 6 Robust Fixture Layout The candidate location points on the datum are generated as shown in Figure 6. 7 Face X Y Z X Nodes 153 81 57 Z Y Figure 6. 7 Candidate... are selected at random 129 Chapter 6 Robust Fixture Layout and their fitness is compared The individual with better fitness is selected as a parent The selection is carried out until the pool size is filled A lower rank and higher crowding distance is the selection criterion The selection is based on the rank and if individuals with the same rank are encountered, the crowding distance is compared Crossover... 130 Chapter 6 Robust Fixture Layout Mutation In the mutation operation, a gene is randomly selected at the point level from a chromosome, and a random point ID is chosen to replace this gene from the node candidates of the surface on which the original point is in order to guarantee the replaced point ID is not out of range The procedure described above is illustrated in Figure 6. 5 The randomly selected... replaced is highlighted with dotted lines and the ID of the randomly chosen point which is to replace point 40 is 98 parent 5 5 5 6 6 3 1 40 65 4 23 55 40 5 5 5 6 6 3 child 1 98 65 4 23 55 98 Figure 6. 5 Genetic operation for mutation In each generation, a population of chromosome is initialized After the reproduction, and the genetic operations of crossover and mutation, the number of the chromosomes... after each iteration cycle 1 36 Chapter 6 Robust Fixture Layout 6. 4 6. 4.1 Experimental Results Experiments An illustrative example is presented in this section to demonstrate the proposed approaches and the test results It is a simplified front knuckle for a generic automotive chassis system The raw part is a casting followed by machining Figure 6. 6 gives the details of the cast and machined parts It is... 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... according to the State Transition Rule (Dorigo et al, 1997) It is directed by both the pheromone amount and the distance from its current location to the targeted ones, ∑ (6. 8) 132 Chapter 6 Robust Fixture Layout t: iteration index : pheromone value between nodes i and j : heuristic distance between nodes i and j : probability to travel from node i to node j : nodes not yet traversed in the ant-tour thus far... (-57.585 -10 .61 2 80.000) (-93.478 -73.005 52. 560 ); (-93.478 -15.271 45.039) (34.148 2.012 37.279) Location face Face X Face Y Face Z Table 6. 7 Objective values Machining Features Face A Face B Orientation (0, 0, -1) (0, 0,-1) Mean location error 1.1E-2 Variation of location error 4.6E-3 Distance of location points 462 .3 Figure 6. 8 Location layout of example part 140 Chapter 6 Robust Fixture Layout 6. 5 Summary . 5 5 5 6 6 3 8 32 93 10 60 8 5 5 5 6 6 3 1 40 65 10 60 8 5 5 5 6 6 3 8 32 93 4 23 55 p arent1 Parent2 child1 child2 5 5 5 6 6 3 1 40 65 4 23 55 5 5 5 6 6 3 8 32 93 10 60 8 . tolerance analysis and robustness. Chapter 6 Robust Fixture Layout 119 6. 2.2 Sources of Localization Errors Three sources of errors, i.e., fixture set-up error, locator profile error and. 1 16 Chapter 6 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