Based on region division setup planning for sheet metal assembly in aviation industry

Abstract

The task of setup planning is to determine the number and sequence of setups and the machining features or operations in each setup. Due to the locating method’s character, the position of the locators may keep away joining tools from arriving at joining points during the joining process of sheet metal. In order to solve these problems, a novel method is proposed, which consists of three parts: (1) the points on the skin are divided into sets of supplied riveting and prior riveting points by convex hull generating method; (2) fixture vector is defined, and a suitable fixture region is generated by vector multiplication; and (3) the particle swarm optimization algorithm is applied to generate optimal setup scheme by setting up the parameters. The new setup planning approach can accomplish riveting work by minimum setup times and workpiece variation without finite element analysis software. At last, a certain kind of aircraft wing joining process is illustrated as an example to demonstrate the effect of the proposed approach.

Keywords

Setup planning supplied riveting set particle swarm optimization algorithm sheet metal assembly

Introduction

Standard computer-aided design (CAD) and computer-aided manufacturing (CAM) tools are now used by industrial organizations to reduce the time and costs of product design and manufacture. It is well known that there is a functional gap between CAD and CAM, which could be bridged by computer-aided process planning (CAPP) only.¹

Setup planning is one of the most important and complex steps in CAPP, which may have a strong impact on manufacturability, total time, and production cost. It has been reported that about 20%–60% machining error is brought by the setup process.² In the aviation, aerospace, and automobile industries, a typical workpiece (such as panel and frame) has the characteristics of large profile and low stiffness, which is susceptible to deformation during the setup and joining process. In this regard, there is a need for the setup planning to feature more locators and clamping devices to maintain the surface. The position distributing and changes of these locators and clamping devices are related to the efficiency and accuracy of the joining assembly work.

Taking panel assembly, for example, the skin and stringer are joined by automatic drilling/riveting system, and the skin is located and clamped on the bracket system as shown in Figure 1. The skin is located by several supporting plates that have the same surface in the contact area such that the position on the skin is in one-to-one correspondence with the supporting plate. When the supporting position on the skin is changed, the supporting plate needs to be taken off and replaced by another corresponding plate. As the positions of the supporting plates change continually throughout the joining process, there is a need to factor this dynamism in the setup planning.

Previous works in the field highlight three methodical approaches: graph theory, knowledge engineering, and intelligent algorithm. Graph theory and knowledge engineering are useful tools for setup and fixture planning. Zhang et al.³ proposed a hybrid graph approach, which can be transferred into directed graph, and tolerance relations are used as critical constraints. Lately, an extended graph describing a feature and tolerance relationship graph (FTG) and a datum and machining feature relationship graph (DMG) were used to support an automobile manufacturing enterprise in fixture design and production planning.⁴ Sun et al.⁵ proposed a new directed graph approach, which could generate many optimal or near-optimal setup plans and provide more flexibility required by different job shops.

More attention is given to intelligence algorithm.^6–8 Ong et al.⁹ presented a concurrent constraint planning methodology and a hybrid genetic algorithm (GA) and simulated annealing (SA) approach for setup planning, and resetup planning in a dynamic workshop environment. Lee et al.¹⁰ used Petri nets as a unified framework for representing both operation planning knowledge and process sequence. Cai et al.¹¹ presented a cross-machine setup planning approach using GAs for machines with different configurations. Alluru and Rao¹² present an application of a newly developed metaheuristic called the ant colony algorithm as a global search technique for the quick identification of the optimal operation sequence by considering various feasibility constrains. Chen et al.¹³ report a new approach to setup planning of prismatic parts using Hopfield neural net coupled with SA.

Some researches have taken tolerance specifications into account. Zhang et al.¹⁴ proposed a graphical approach to generate optimal setup plans based on design tolerance specifications. Bansal et al.¹⁵ developed an integrated setup and fixture planning system for minimizing tolerances at critical regions using a data exchanged part model as an input. Huang and Liu¹⁶ provided a tolerance normalization approach for direct comparison of different types of geometric tolerances and dimensional tolerances. Yao and Han¹⁷ developed an automated setup planning technique and system based not only on the tolerance analysis but also the manufacturing resource capability analysis. Kang et al.¹⁸ presented a new approach for fixture tolerance analysis that is more generalized and can be used to assign locator tolerances based on machining surface tolerance requirements.

All these methods provide a strong basis for setup planning. However, the main problem of knowledge-based and graph-based approaches is that there is no consideration with the precedence constraints during setup formation. The constraint of tolerance cannot be neglected for product quality. On the other hand, due to the demand of supplied riveting, the problem in this study is a special dynamic planning that needs to find a new approach to solve the problem. Intelligence algorithm can be applied to accelerate planning.

In summary, there are three aspects to this problem that have to be addressed. The first problem relating to the layout of supporting plates is whether all the joining points could be riveted. The second aspect deals with the variation of the workpiece, which may impact the position accuracy of the joining points on the skin. The third question involves locating times that are intimately related to assembly efficiency. Based on the review, setup planning for sheet metal assembly needs to consider compliment rates, locating times, and the variation of the workpiece.

In this article, a new setup planning method is proposed on the basis of the supplementary riveting (SR) region definition. First, the points around fixtures are divided into sets of Supplied Riveting (SR) and Prior Riveting (PR) points using convex hull. Because the joining tools are obstructed by supporting plates, clamps, or other fixed components, points in SR sets cannot be riveted. In CSR sets, the points can be riveted preferentially. Second, the fixture vector is defined and a suitable fixture region is generated by vector multiplication. Besides, moving steps $Δ x_{i}$ and $Δ {\hat{x}}_{i}$ and the distance $d$ need to be confirmed. Finally, the particle swarm optimization (PSO) algorithm is applied to generate optimal fixture planning by setting parameters, which aimed at maximizing the number of riveting points in this setup, and minimizing the setup times and workpiece deformation.

Principles and constraints for setup planning

The research of fixture method is not only applied to part machining field but also is used in product assembly process widely. Generally, the relative position among subassemblies is fixed by locators and temporary connectors, which might have the property of disassembly and recycling, such as bolts. Then, the subassemblies are connected by riveting or welding technology. So the fixture may prevent welding gun or riveting hammer from the position of joining points and affect assembling efficiency.

The positioning method of sheet metal is different from the box-shaped workpiece. For box-shaped workpiece, the most general method is the “3-2-1” positioning criterion proposed by Asada in 1985.¹⁹ By selecting three mutually perpendicular planes, the foundation of “3-2-1” positioning criterion is established, which is composed of principal reference, secondary reference, and third reference, as shown in Figure 2. The important surface of the workpiece may be generally selected as principal datum. The criterion constrains six freedoms of the workpiece and considers the requirement of design and assembly simultaneously.

According to Fig. 2, plane A is the principal reference, and the locators lie on the characters that can be represented as L₁, L₂ and L₃, which control a translation T_z along Z axis and two rotation R_x and R_y around X and Y axis respectively. Then plane B is selected as secondary reference, and two locators lie on the character L₄ and L₅. So the translation T_y along y axis and rotation R_z around z axis are controlled. Finally, the third reference C is defined by character L₆, and translation T_x along X axis is constrained. Through controlling the three planes A, B and C the workpiece is constrained completely^[18].

For sheet metal, the demand must be improved. On one hand, in aviation industry, sheet metal is generally used for the skin in wall panel of the wing and fuselage, which is made of aluminum alloy, and the curvature is one of the most important parameter for aerodynamic performance. On the other hand, due to the material and the structure, sheet metal has the character of low stiffness. Thus, the curvature may be deformed during assembly process for the reason of clamping and joining forces.

Therefore, during assembly process, clamping needs to achieve two aims: one is to maintain the profile of workpiece, and the other is to protect workpiece and frock from relative movement when joining force is applied to the workpiece. In the principal datum, the number of locators may be hard to satisfy this requirement. So the “N-2-1” positioning criterion for sheet metal can be adopted.

It means that N locators are used for positioning and supporting the workpiece in principle reference. The workpiece is in an overconstraint condition. The positioning scheme of the skin is shown in Figure 4, which is also used in other sheet metal assembly positioning.

After positioning the sheet metal completely, the clamping operation can be carried out. For reducing the clamping deformation, the clamping force can be applied on the symmetric position of the locators. The direction of supporting force is contrary to that of clamping force. So, it is important to pay special attention to the position distribution of the locators. By setup planning, the position of joining points can be assured to lie on the ideal positions, which assures that the assembly work will be launched smoothly.

Setup planning method

For maintaining the surface of skin, N locators are scattered on the surface of the workpiece, which may separate joining tools from the joining points with the position of the locators. So the target of setup planning is to accomplish riveting work by minimum setup times and minimum variation of the workpiece. The method is represented as follows.

Segment definition for fixture

Clamping force acts on the feature surface of the workpiece generally. For some features, such as chamfers and circular conical surfaces, it is difficult to load force and moment. So segments suitable for fixture need to be defined.

Define the distance of fixture by selecting two points $p_{0}$ and $p_{n}$ from the boundary of the skin and generate a line $F$ between the two points. Let the line $F$ be located at the surface of the skin, which ensures that the following work is launched on the surface of skin. Now the line can be called the space of fixture.

Then, insert $b$ points $p_{1}, p_{2}, \dots p_{b}$ on the line, after which the space of fixture is divided into $(b + 1)$ segments. Each segment corresponds to $f_{i}$ , and the space of fixture is expressed as $F = {f_{1}, f_{2}, \dots, f_{b + 1}} = {(p_{0}, p_{1}), (p_{1}, p_{2}), \dots, (p_{b}, p_{n})}$ .

The vector is defined as $K = (k_{1}, k_{2}, \dots, k_{b + 1})$ , where $k_{i}$ corresponds to $f_{i}$ . When

k_{i} = {\begin{matrix} 1 & f_{i} suitable for fixture \\ 0 & f_{i} not suitable for fixture \end{matrix} i = 1, 2, \dots, b + 1

So the segments $F'$ suitable for fixture is obtained by $K^{T} F$

K^{T} F = (\begin{matrix} k_{0} \\ k_{1} \\ ⋮ \\ k_{b} \end{matrix}) (\begin{matrix} (p_{0}, p_{1}) (p_{1}, p_{2}) \dots (p_{b}, p_{n}) \end{matrix}) = {(\begin{matrix} k_{0} • (p_{0}, p_{1}) k_{1} • (p_{1}, p_{2}) \dots k_{b} • (p_{b}, p_{n}) \end{matrix})}^{T}

SR region division

Suppose there are $n$ riveting points $R = {r_{1}, r_{2}, \dots, r_{n}}$ on the skin, then the skin and the supporting plates are located on the frame by $m$ locators $L = {l_{1}, l_{2}, \dots, l_{m}}$ . Now the definition of $SR$ is given.

Definition 1

$SR$ set $S_{L_{j}}^{o} = {\forall r_{i} | | r_{i} - l_{j} | \leq d r_{i} \in R}$ . Where $o$ expresses the times of setup, $i = 1, 2, \dots, n$ , $j = 1, 2, \dots, m$ .

It means when the skin is clamped by oth time, $SR$ set $S_{L_{j}}^{o}$ contains the riveting point $R_{i}$ and that the distance between the riveting point and the locator is lesser than the given distance $d$ . Based on the above-mentioned analysis, the locator may be the hole/pin, the block, supporting plates, and so on. If the locator is a hole/pin or a block, the locator can be simplified to a point, and the center of the locator becomes the position of the point. If the locator is a supporting plate, the locator can be simplified to a curve on the surface. And the position of the curve comes from the half of the supporting plate’s width.

The distance $d$ related to the diameter of riveting head of the automatic drilling/riveting machine, the thickness of supporting plate, and other factors, which may keep away the riveting head from the riveting point. In addition, the number of sets is generally same as the number of positioning and clamping elements. When

S_{L_{j}}^{o} = {\begin{matrix} ϕ unexist points satisfy the requirment among l_{j} \\ \neg ϕ exist points satisfy the requirment among l_{j} a \geq 1 \end{matrix}

Given the position of locators and the distance $d$ , the obstructed riveting points are put in a single set $S_{L_{j}}^{o}$ , and on the other hand, the complementary set of the $SR$ set is the riveting points that can be riveted in this clamping method, which is named as the $PR$ set, and is shown in Figure 5

{\bar{S}}^{o} = C_{R} (\cup S_{L_{j}}^{o}) = {\forall r_{i} | r_{i} \notin \cup S_{L_{j}}^{o} r_{i} \in R} i = 1, \dots, n j = 1, \dots, m o \geq 1

Definition 2

$SR$ region. the riveting points in the $SR$ set are connected, and the formed convex hull $A_{L_{j}}^{o}$ is named as the $SR$ region. The $A_{L_{j}}^{o}$ is constructed by linear combination of all the riveting points in $R$

A_{L_{j}}^{o} = {\sum_{k = 1}^{a} t_{k} r_{k} | r_{k} \in S_{L_{j}}^{o}, \sum_{k = 1}^{a} t_{k} = 1, t_{k} \in [0, 1]} j = 1, \dots, m

Region $SR$ is called as convex if it satisfies that the segment $r_{1} r_{2}$ entirely belongs to $SR$ for any two points $r_{1}$ and $r_{2}$ belonging to $SR$ . There are many different convex hull algorithms, for example, Granham scan algorithm,²¹ Andrew algorithm,²² Quickhull algorithm,²³ and Datta and Parui’s algorithm.²⁴ All the above-mentioned algorithms are widely used, but Quickhull algorithm and Andrew algorithm have been considered the two best algorithms in a latest review of convex hull algorithms.²⁴ Recently, there are also some studies on convex hull algorithm.^25–28 So, there is no further discussion about convex hull algorithm in this article.

Through the definition, the surface of the skin is divided into multiple regions. As shown in Figure 6, riveting points complementally exist in $SR$ region $A_{L_{j}}^{o}$ , and the others are the priority riveting points, which constitute $PR$ region. Then the following work is based on the divided regions.

There are generally several setups in an assembly process. For each setup, some of the locators have fixed position, and others may be moved. The fixed ones can be checked for interference with joining tools. The moving ones are the key research objects in this article that may impact the times of setup, and it is known as supporting plate generally for skin of aircraft.

Setup planning based on SR region

For the dynamic fixture problem, the most important thing is to find the initial position of the supporting plate, which may impact the number of supplied riveting points. When a workpiece is set up at the beginning, the times of the setup must be considered, which may lead to the increase of deformation rate, and reduce efficiency and accuracy of the product ultimately. So the setup planning method based on the $SR$ region is proposed as follows.

First, record the initial position of the supporting plate. Let the position of $p_{0}$ be 0, and $| p_{0} p_{n} |$ be the distance of fixture space, denoted as $d_{1}$ . Record the coordinates of two endpoints in the supporting plate. Then, the initial position is $l_{j}^{o} = {l_{1}^{o}, l_{2}^{o}, \dots, l_{m}^{o}}$ , as shown in Figure 7. $j$ is the serial number of supporting plates, and $o$ is the times of setups. Certainly, the initial position belonging to the segments is suitable for fixture. For the other endpoint, a similar initial position, ${\hat{l}}_{j}^{o} = {{\hat{l}}_{1}^{o}, {\hat{l}}_{2}^{o}, \dots, {\hat{l}}_{m}^{o}}$ , is generated.

Second, the two endpoints of supporting plate are moved by step lengths $Δ x_{i}$ and $Δ {\hat{x}}_{i}$ . If the endpoint moves to the segment that might not be suitable for fixture, continue moving until the endpoint leaves this segment or reaches $p_{n}$ . Place the position coordinates of one endpoint in one set $g^{o} = {l_{1}^{o}, l_{2}^{o}, \dots, l_{x}^{o}}$ and the others in another set ${\hat{g}}^{o} = {{\hat{l}}_{1}^{o}, {\hat{l}}_{2}^{o}, \dots, {\hat{l}}_{y}^{o}}$ . The coordinates are selected from the two alternative sets, respectively; thus, a position of a supporting plate can be obtained. In this way, a group of supporting plate positions has been gained, marked as $q^{o} = {q_{1}, q_{2}, \dots, q_{a}}$ . Where $a$ is the number of positions in this group.

Third, give the distance $d$ and generate $SR$ set corresponding to ${q_{1}, q_{2}, \dots, q_{a}}$ . Generate $SR$ set aimed at one supporting plate $l_{j}$ in the first setup. Then generate $SR$ set $S_{l_{j}}^{o}, j = 1, 2, \dots, m$ aimed at $l_{j}$ in the $oth$ setup, where $m$ is the number of supporting plates. So the number of sets can be gained by doubling the number of supporting plates, as shown in Figure 8.

We can obtain the union, $S^{o}$ , of all the sets, $S_{L_{j}}^{o}$ , for the oth setup. For example, $S^{1}$ is the union set of $S_{l_{1}}^{1}, S_{l_{2}}^{1}, \dots, S_{l_{m}}^{1}$ . In the same way, $S^{1}, S^{2}, \dots, S^{a}$ is obtained. Therefore, the compensation set of $S^{a}$ is defined as ${\bar{S}}^{a}$ , which includes the riveting points in every setup. The process is described in Figure 9.

Finally, the question is transformed into a combinatorial optimization problem. The constraint condition is that all the points can be riveted. The PSO algorithm is proposed to generate optimal fixture planning.

The PSO algorithm is an evolutionary computation method introduced by Kennedy and Eberhart.²⁰ The velocity is constantly updated according to the particle’s own experience and the experience of the whole group of particles. The best experiences of the particle itself and its neighbors are memorized for updating the velocity and position. For finding the optimal solution, each particle $i$ updates its velocity $v_{i}^{k + 1}$ according to the following velocity updating equation and the position updating equation

v_{i}^{k + 1} = {wv}_{i}^{k} + ξ c_{1} (P_{i} - x_{i}) + η c_{2} (P_{g} - x_{i})

(1)

where $v_{i}^{k}$ and $v_{i}^{k + 1}$ are the velocity of the particle $i$ in the current iteration k and next iteration $(k + 1)$ ; $w$ is called the inertia weight; $ξ$ and $η$ are the independent random numbers with a uniform distribution [0, 1]; $c_{1}$ and $c_{2}$ are constants called acceleration coefficients weight, which expresses how much the particle uses its own experience and swarm’s past experience, respectively; $P_{i}$ is the best position of each individual particle $i$ ; $P_{g}$ is the best position of the entire swarm; and $x_{i}$ is the position of a particle $i$ .

With the updated velocity, the position of each particle is updated with the following equation

x_{i}^{k + 1} = x_{i}^{k} + v_{i}^{k + 1}

(2)

where $x_{i}^{k + 1}$ is the position of the new particle in the iteration $(k + 1)$ , $x_{i}^{k}$ is the position of current particle in the iteration k, and $v_{i}^{k + 1}$ is the velocity of the particle.

The fitness function is constructed as follows

F (t_{i}, d_{i}, r_{i}) = ω_{1} f_{u_{1}} (t_{i}) + ω_{2} f_{u_{2}} (d_{i}) + ω_{3} f_{u_{3}} (r_{i})

(3)

where $ω_{1}, ω_{2}, and ω_{3}$ are the weights, and $ω_{1} + ω_{2} + ω_{3} = 1$ ; $f_{u_{1}} (o_{i})$ is the radio of fixture times; and t_i is the fixture times. The ratio increases with the time. Generally, it needs to move twice to finish the joining work. When $o_{i} = 2$ , $f_{u_{1}} (o_{i}) = 0$

f_{u_{1}} (o_{i}) = \frac{(o_{i} - 2)}{o_{i}}

$f_{u_{2}} (d_{i})$ refers to the variation radio of workpiece. It is an important index that immediately relates to the joining accuracy. As the distance between riveting points and the nearest supporting plate increases the effect of gravity becomes deeper. On the other hand, as the distance between riveting points and the nearest clamping point is furthered, the deformation effect of the clamping force is smaller. d_i is the distance between the riveting point and the nearest supporting plates, and d′ is the distance between the riveting point and the nearest clamping points

f_{u_{2}} (d_{i}) = \frac{\sum_{i = 1}^{n'} d_{i}}{n' D} + \frac{\sum_{i = 1}^{n'} {d'}_{i}}{n' D'}

where D is the maximum distance between the two supporting points. D′ is the farthest distance between the rivet point and the clamping point, and n′ is the number in the PR set.

$f_{u_{3}} (r_{i})$ expresses the riveting radio for the first time. r_i is the number in the PR set for the first setup

f_{u_{3}} (r_{i}) = \frac{(n - r_{i})}{n}

where n is the total number of riveting points on the skin.

The number of particles is related to the searching time and the solution quality. The detailed steps are expressed in the following flowchart (Figure 10).

This setup planning method can help engineers find the appropriate initial position of supporting plates rapidly; it only needs input of some parameters. The optimal output includes the position of every supporting plate and the times of setup.

Example

In order to demonstrate the effectiveness and robustness of the setup planning method, a wing panel is chosen as an example, which is made up of a skin and 11 stringers. The CAD model of the skin is shown in Figure 11.

$p_{0}$ and $p_{n}$ are selected, and a line $F$ is generated on the surface. Eight points, $p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}, p_{7}, and p_{8}$ , are inserted, and the line is divided into nine segments. The coordinates of these points are given in Table 1.

Table 1.

Ideal coordinate.

ID	X (mm)	Y (mm)	Z (mm)	ID	X (mm)	Y (mm)	Z (mm)
$p_{0}$	30211.1	−12193.2	304.3	$p_{5}$	30445.16	−12208.1	304.3
$p_{1}$	30259.144	−12196.268	304.3	$p_{6}$	30491.97	−12211.1	304.3
$p_{2}$	30304.722	−12199.178	304.3	$p_{7}$	30538.78	−122214.1	304.3
$p_{3}$	30351.533	−12202.167	304.3	$p_{8}$	30585.59	−12217.1	304.3
$p_{4}$	30398.344	−12205.156	304.3	$p_{n}$	30632.4	−12220.1	304.3

Define a vector $K = (1, 0, 1, 1, 0, 1, 1, 0, 1)$ . The segments $F'$ is obtained by $K^{T} F$

F' = K^{T} F = (\begin{matrix} 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \end{matrix}) ((p_{0}, p_{1}) (p_{1}, p_{2}) (p_{2}, p_{3}) (p_{3}, p_{4}) (p_{4}, p_{5}) (p_{5}, p_{6}) (p_{6}, p_{7}) (p_{7}, p_{8}) (p_{8}, p_{n})) = (\begin{matrix} (p_{0}, p_{1}) (p_{2}, p_{3}) (p_{3}, p_{4}) (p_{5}, p_{6}) (p_{6}, p_{7}) (p_{8}, p_{n}) \end{matrix})

The six intervals $\begin{matrix} (p_{0}, p_{1}) (p_{2}, p_{3}) (p_{3}, p_{4}) (p_{5}, p_{6}) (p_{6}, p_{7}) (p_{8}, p_{n}) \end{matrix}$ are suitable for fixture. The panel with fixture intervals is shown in Figure 12. After setting $p_{0} = 0$ , the distances between each points are shown in Figure 13.

The distance between $p_{0}$ and $p_{n}$ is about 4.2 m, so four supporting plates $L_{1}$ , $L_{2}$ , $L_{3}$ , and $L_{4}$ are adopted to locate the constrain freedom in $z$ direction. $L_{5}$ , $L_{6}$ , and $L_{7}$ constrain the freedom in $x$ direction and $y$ direction. First, put the supporting plates at the position of $p_{0}$ , $p_{2}$ , $p_{4}$ , $p_{5}$ , and $p_{8}$ , as shown in Figure 14.

Define $d = 7 mm$ , then the $SR$ set is counted as shown

\begin{matrix} S_{L_{1}} = Φ, S_{L_{2}} = {r_{192}, r_{193}, \dots, r_{214}}, \\ S_{L_{3}} = {r_{582}, r_{583}, \dots, r_{601}}, \\ S_{L_{4}} = {r_{783}, r_{784}, \dots, r_{797}}, S_{L_{5}} = Φ, S_{L_{6}} = Φ, \\ S_{L_{7}} = Φ \end{matrix}

The left riveting points are put into ${\bar{S}}^{a}$ , where $a$ is the time of fixture. Set the moving step length $Δ x_{i} = 5, i = 1, 2, 3, 4, 5$

{\bar{S}}^{a} = {\bar{S}}^{1} = {r_{1}, r_{2}, \dots, r_{191}, r_{215}, \dots, r_{392}, r_{415}, \dots, r_{581}, r_{602}, \dots, r_{782}, r_{798}, \dots, r_{865}}

Let these conditions be the input conditions. Set up initial parameters $ω_{1} = 0.3, ω_{2} = 0.3, ω_{3} = 0.4$ , $N = 100$ ; inertia weight $w = 0.9$ , $c_{1} = c_{2} = 2$ ; and random number $ξ = η = 0.5$ . The value of optimal fitness function converges in the iteration step 310 and stabilizes on 0.07. The plot is shown in Figure 15.

By running the PSO optimization program, the setup scheme has been outputted. By setting up twice, all the points are riveted, as shown in Figure 16.

The setup schemes are shown in Table 2. The times of setup accords with the practical situation, and the detailed position can be applied to the assembly process directly.

Table 2.

Detail scheme.

Support plates	X (mm)	Y (mm)	Z (mm)	Support plates	X (mm)	Y (mm)	Z (mm)
$P_{L_{1}}^{1}$	30223.1	−12194.5	304.3	$P_{L_{1}}^{2}$	30235.1	−12195.1	304.3
$P_{L_{2}}^{1}$	30340.7	−12202.2	304.3	$P_{L_{2}}^{2}$	30364.7	−12202.9	304.3
$P_{L_{3}}^{1}$	30469.2	−12212.9	304.3	$P_{L_{3}}^{2}$	30475.2	−12213.4	304.3
$P_{L_{4}}^{1}$	30597.6	−12217.7	304.3	$P_{L_{4}}^{2}$	30609.6	−12218.1	304.3

Conclusion

A novel setup planning method aimed at the problems in the assembly of sheet metal is proposed, which can solve dynamic planning problem of supporting points or clamping points, considering the variation radio of workpiece and the riveting radio at the same time. Especially for sheet metal, the number of supporting points is more than in the other parts for maintaining the surface, and the position is dynamic, which needs to change in the process of joining assembly for avoiding blocking the joining tools. The PSO algorithm is used to improve calculation efficiency of the combinatorial optimization.

This method also can extend to setup planning for other types of locators, by modification of the coverage of d. In addition, the method can be used for situation optimization of fixture points without finite element analysis (FEA) software. Future work will consider a more accurate expression of the workpiece variation, and will also need the addition of clamping force to the setup planning.

Footnotes

Acknowledgements

We would like to thank the editors and the anonymous referees for their insightful comments.

Funding

This study was supported by National Natural Science Foundation of China (50805119), Aeronautical Science Foundation of China (2010ZE53030), and the Foundation for Basic Research of Northwestern Polytechnical University, China (JC201032).

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