Modelling and layout design for an automated fibre placement mechanism

Mechanism and Machine Theory 144 (2020) 103651

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Modelling and layout design for an automated fibre placement mechanism Wuxiang Zhang a,b, Fei Liu a,b,∗, Yongxin Lv a,b, Xilun Ding a,b a b

School of Mechanical Engineering and Automation, Beihang University, 37 Xueyuan Road, Beijing 100191, China Ningbo Institute of Technology, Beihang University, Ningbo 315832, China

a r t i c l e

i n f o

Article history: Received 27 July 2019 Revised 21 September 2019 Accepted 1 October 2019

Keywords: Automated fibre placement Fibre transmission path Coordinate transformation Layout design optimization

a b s t r a c t The fibre transmission path in an automated fibre placement (AFP) mechanism greatly affects the final embedded defects and mechanical properties when producing composite components. In this study, the technical requirements, logical movements and complex coupled relationships within multiple functional modules during fibre transmission are analysed. Using the functional classification method, we decompose AFP mechanism into four subsystems: the creel assembly, the tension measurement unit, the CCR unit (clamping cutting and restarting) and the compacting and heating mechanisms. To express the orientation and position of each subsystem as design variables, coordinate transformation is introduced. On the basis of the multidisciplinary optimization principle and the complex coupled relationship within the AFP mechanism, a multi-objective layout optimization with nested analysis and design approach is proposed for each functional component to address the complex mechanism layout problem. Finally, a 4-strip and 6-strip AFP mechanisms are then assessed to verify the proposed methodology. The results demonstrate that this approach can effectively realize the layout design for the AFP. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Automated fibre placement (AFP), which is capable of producing composite materials parts, has become the main automatic manufacturing technology in aerospace and other industries [1]. During the fibre placement process, a number of separated resin-impregnated fibre tows from the creel assembly will be fed at independent rates through the above subsystems and finally collimate into a fibre band on the surface of the mandrel [2–4]. From the early 1980s, industrial companies and institutions have made tremendous effort in AFP structural design to achieve the specific functional requirements, such as Boeing Company [5,6], Cincinnati Machine (CM) [7] and ADC Acquisition Company [3]. Currently, the methodologies of metamorphic mechanisms and configuration synthesis [8–13] have been proposed to overcome the limitation of traditional mechanisms and adopted in AFP mechanical design [14,15]. Then more attention has been paid to the AFP process optimization and defect analysis for improving fibre performs quality. Experiments and studies [16–19] have systematically shown the relationship between the fibre path defects and mechanical properties, and concluded that fibre path defects have destructive impacts on the resin flow and characteristics such as the shear loading and compressive strength. Since the prepreg will inevitably bend and twist repeatedly and slip to one side of the guide roller when tows are threaded through the functional subsystems, inner defects such as fractured into fibre bundles or overlap and lap will be made. Further, there are ∗

Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (F. Liu), [email protected] (Y. Lv), [email protected] (X. Ding).

https://doi.org/10.1016/j.mechmachtheory.2019.103651 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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W. Zhang, F. Liu and Y. Lv et al. / Mechanism and Machine Theory 144 (2020) 103651

complex constraints and strong coupling among these functional subsystems, these characteristics determine that the layout design for each functional module can promisingly optimise the fibre transmission path and mitigates the defects. The layout design has arisen and been studied in the field of the travelling salesman problem [20–23], bin packing problem [24–27], scheduling problem [28,29] and clustering problem [30,31] etc., which can be regarded as a combinatorial optimization or non-deterministic polynomial hard (NP-hard) problem [32]. The traditional layout optimization algorithms are ineffective and usually obtain a local optimal solution. Currently, heuristic algorithms, however, are the better options [27]. Among them, differential evolution (DE) algorithm is the one of the most efficient optimiser [33,34] .Wang et al. [35] applied the divide-conquer-coordination strategy in satellite modules to decompose the SMLDP into several layout sub-problems. Liu and Teng [34] presented a multiobjective optimization model of the cutter layout integrating expert experience and an evolutionary algorithm, which advantageously reduces or replaces the complexity of the method. Teng et al. [36] then proposed a dual-system framework for satellite modules that increased the diversity of the population and decreased the premature convergence. In previously proposed AFP optimization methods, the layout design within AFP mechanism was rarely mentioned. The traditional AFP mechanism layout design schemes [37–39] can greatly contribute to the reduction of the costs during the machine manufacturing and maintenance in the aerospace composite industries. However, these schemes will inevitably induce fibre defects and affect the layup precision. Herein, we emphasize on the fibre transmission path and mitigating fibre damage during the AFP mechanism layout design, and present a layout optimization strategy which is different from the traditional engineering solutions. The achievement of this study can provide theoretic guide for more variety of automation design which has characteristics of multiple degrees of freedom, multiple drives, complex constraints and strong coupling, and promote its development and application. The structure of the paper is organised as follow. Section 2 analyses the fibre transmission path, and obtains the layout technical requirements. Section 3 introduces a novel optimization strategy and a process to solve the layout problem of such a complex system. Based on the optimization strategy, a mathematic model is established for the AFP system in Section 4. Section 5 shows that the 4-strip and 6-strip AFP mechanisms are optimized using the differential evolution algorithm, and the results obtained verify the optimization strategy. This study claims that this attempt would optimise the AFP layout design and be promising to reduce the prepreg tow’s physical damage and inner defects of performs. 2. AFP layout technical requirements The AFP process is mostly derived from the machine tool concept [40,41], which was developed to overcome the disadvantages of winding machines, manual lay-ups and automated tape laying machines. A typical AFP system includes a fibre placement head and an industrial robot (usually supplied by ABB or KUKA Company) or gantry machine, which satisfies the requirements for flexible tool movement in larger components manufacturing and improves the consistency of the production quality. Fig. 1 shows the main AFP systems from the Electroimpact Company, M. Torres and Coriolis [3,37,42]. The fibre placement head is designed on a machine carriage and can be individually disassembled with respect to the creel assembly, tension measurement unit, CCR unit (clamp, cut and restart mechanism), and compacting and heating mechanism, resulting in the precise control of the moulding parameters. To analyse the layout and technical requirements of the AFP, an illustration of a single fibre transmission path is presented, as seen in Fig. 2. The tow is first pulled from the creel assembly and goes through the guide roller, tension measurement unit, and CCR unit. Eventually, the fibre tow is laid onto the mould surface and consolidated by the compacting and heating mechanism [14]. As shown in Fig. 2, the creel assembly includes pairs of rotating spools that continuously supply the tows that are applied. Meanwhile, the servo motor is the driving device that is connected to the shafts of the rotating spools, providing a stable

Fig. 1. AFP systems from the Electroimpact Company, M. Torres and Coriolis [3,37,42].

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Fig. 2. Process analysis for a single fibre tow.

velocity and adjusting the tension force. There are two ways to arrange the creel assembly. In some cases a creel assembly is separately mounted on the cradle and redirect mechanism is provided between the assembly and AFP head, which is traditionally used for more than 16 fibre tows. This would result in high lay-up speed and precision because of weight reduction of AFP head, but involve the complicated maintenance and fibre guidance, for example, [33] designed a servo driven redirect roller mechanism within fibre placement system to maintain tension and position of fibre. Besides there is no fibre path optimization problem in this arrangement. Then other researchers installed the creel assembly on the AFP head to shorten the transmission path and it can quickly interchange modular AFP mechanisms [37] as shown in Fig. 1(a) and (b). The guide rollers, as auxiliary sections, are added to redirect the fibre tows. In this study, we will discuss how to build and layout the latter configuration for fibre transmission path optimization. The tension measurement unit consists of a force sensor and a roller that measure the fibre tension, which has a great influence on the production quality. The tension control provides the necessary tension for the tows and compensates for the different dynamic behaviour of the AFP by controlling the servo motor of the creel assembly. The CCR unit enables the AFP mechanism to drop or add tows, which can make a flexible adjustment for the fibre bandwidth [24,25]. While the CCR unit is at work, the tow is first clamped and then cut off by a knife. While the tow is fed to broaden the bandwidth, under the action of the restarting shaft, the restarting wheel would feed the tow to the end of the AFP along with the simultaneous withdrawal of the knife and clamping block. Since the movements of the restarting shaft in each CCR unit are the same, the restarting shafts can be integrated according to the layout design. After the CCR unit, all the fibre tows are finally transported to the compacting and heating mechanism that exerts a compressive force and provides an appropriate temperature for the prepreg curing. The compacting motor, force sensor and heat device are the indispensable parts of the composite consolidation. Since the positions of the compacting and heating mechanisms are relatively independent of the other relevant factors, the layout design of this module and other auxiliary subsystems can be easily determined after the positions of the main subsystems are given, which are not included in AFP mechanism layout optimization. 3. Problem statement of layout optimisation Based on the above analysis, there exist complex connections and constraints between each section within the AFP mechanism. Therefore to clearly formulate the problem, the AFP configurations can be treated as the cylindrical modules according to Fig. 1(a) and (b) and coordinate transformation is introduced to illustrate the position and orientation of these layout objects. Three bearing plates (S1 –S3 ) attached to the module are used to install these subsystems. Specifically, coordinate system Oxyz is the reference coordinate system. CCR units are located on the S3 bearing plate and the compacting and heating mechanism are located with the fibre transmission channel. Tension measurement units are located on the S2 bearing plate and coordinate system O’x’y’z’ is parallel to the reference coordinated system. Creel assemblies are located on the front of the bearing plate S1 as shown in Fig. 3. Taking the configuration in Fig. 3(a) as the object to be optimised, the modelling method and optimization process for Fig. 3(b) are similar to the previous one. In every bearing plate, the fibre channel is located on each bearing plate S1- S3 so that the fibre can be threaded through the plate. Generally, the fibre channel space on the bearing plate S1 was modelled as a circle. But in order to simplify the complex of modelling and interference computations, the following two on the bearing plates S2 , S3 are hexagons. Besides, after the layout design of the AFP, its size and shape can be slightly adjusted by engineers combined with actual conditions. When the above mechanisms work and act together, the height of each subsystem is nearly invariable; hence, the layout design of each bearing plate can be simplified to a two-dimensional (2D) problem. To reflect geometric characteristics of each subsystem, these subsystems are then replaced by two-dimensional irregular polygons or circles, which is similar to a bounding box. The centroid of each subsystem can be given by a three-dimensional model from the SolidWorks system. As mentioned before, the fibre tow is first pulled from the creel assembly so that the creel assembly is attached to the front side of upper bearing plate S1 . Fig. 4(a) shows a pair of creel assemblies in the two-dimensional space, and a servo motor is mounted on the reverse side of upper bearing plate S1 . Xi is the centre of the creel assembly in the coordinate systemX O Y . r is the maximum radius of the creel assembly. X i is defined as an output point of creel assembly where

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Fig. 3. Two types of the AFP mechanism structure.

Fig. 4. The definition of an object on a bearing surface.

the fibre tow will pass. Once the position matrix of the creel assembly is determined, the output point can be determined from the geometric relationship. A reserved circular space with a radius of r connects with the robot and serves as a fibre channel. The position matrix of the creel assembly can be described by using a transformation matrix: 0

0 Xi = Trans ⎡ (xi , yi , 0)Trans ⎤⎡(0, 0, z2 ) X 1 0 0 xi 1 0 0 ⎢0 1 0 yi ⎥⎢0 1 0 =⎣ 0 0 1 0 ⎦⎣0 0 1 0 0 0 1 0 0 0

⎤⎡

0 1 0 ⎥⎢0 z2 ⎦⎣0 1 0

0 1 0 0

0 0 1 0

⎤

0 0⎥ 0⎦ 1

(1)

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where z2 represents the height of the creel assembly centre in coordinate system Oxyz , which is a constant. 0 X is a unit matrix in the coordinate system Oxyz . Similar to the previous plate, the tension measurement unit is attached to the front side of middle bearing plate S2 . As shown in Fig. 4(b), it could be modelled as a convex polygon. Pik is defined as a vertex of a polygon in the coordinate system X O Y . nik is the unit normal vector for each edge of the convex polygon. Mi is the centre of the tension measurement unit −−−→ in the coordinate system X O Y . ui is a unit vector along Mi Mi . M’i is defined as an output point through which the fibre tow will pass, and hexagon C is reserved for the fibre channel. Ci is vertex of the hexagon, and ki is the unit normal vector of each edge of hexagon C. The position matrix of the tension measurement unit can be represented as follows: 0

Mi = Trans(mi , ni , 0 )Rot(Z, θ )Trans(0, 0, z1 )0 X

(2)

where θ is the angle between unit vector ui and the X -axis. z1 is a constant, denoting the height of the tension measurement unit centre in coordinate system Oxyz . 0 X is a unit matrix in coordinate system Oxyz . The CCR unit is attached to the reverse surface of lower bearing plate S3 . As shown in Fig. 4(c), the CCR unit could be modelled as a convex polygon in the two-dimensional plane. Qik is defined as a vertex of a polygon in coordinate system XOY. tik is a unit normal vector for each edge of the convex polygon in coordinate system XOY. Wi is the centre of the CCR unit in the coordinate system XOY. W i is defined as an output point through which the fibre tow will pass. vi is a unit vector along the point Wi to one of the vertexes. The hexagon is reserved for the fibre channel and restarting shaft. The position matrix of the subsystem can be represented as follows: 0

Wi = Trans(wi , ei , 0 )Rot(Z, γ )Trans(0, 0, z0 )0 X

(3)

where γ is the angle between unit vector vi and the X-axis. z0 is a constant denoting the height of the CCR unit centre in coordinate system Oxyz . Hence, the AFP layout parameters can be described as n

X = ∪ {xi , yi , mi , ni , θi , wi , ei , γi } i=1

(4)

where n is the number of fibre tows. Thus, the design objective here is to optimise the fibre path of bending angle, length and occupied space of the whole AFP system (the smaller the better) and also to satisfy the following constraints: (1) All allocated modules should be contained within the each bearing plate, with no clashing among the modules; (2) There should be no overlapping between the allocated modules and fibre channel; (3) Each fibre path should not interfere with each other. The detailed modelling process and the mathematical model are described in next section. 4. Layout design of the AFP 4.1. Layout design method According to above analysis, the AFP mechanism is highly integrated, and it has the characteristic of multiple connections between each functional module. The traditional layout design decomposes the system into several subsystems; however, the layout design of the AFP must ensure the constraints in each bearing plate and take into account the connections and constraints between the relative bearing plates in the global optimization. For the purpose of achieving global optimal results, the independent variables, objective functions and constraints of the objects that are attached to each bearing plate of the model should be first obtained. The global objective functions and constraints between the adjacent plates are described next, and the integral optimization model is then established. The establishment and solution process of the global optimization model can be expressed as a flow chart, as shown in Fig. 5. The specific optimization process is illustrated as follows. (1) System analysis and layout technical requirements First, the structure of the system should be analysed, and the layout technical requirements, logical movements and complex coupled relationships can be concluded based on the above analysis. Furthermore, the definition of the system is determined and all variables of the layout design can consequently be obtained. (2) Establishment of optimization model for each subsystem level and integral system level Second, the original coupled system can be decomposed into several relatively independent subsystems. In each subsystem, the optimization objects and constraints can be established based on the required performance. The relationships between the adjacent subsystems are summarised herein so that the independent variables of the layout design can be determined from all variables and the rest of the objectives and constraints can lastly be obtained. After the analysis, the global optimization model of the system is eventually established. (3) Solution and optimised results for the model After the global optimization model is built, a multidisciplinary design optimization approach [43–46] which can solve design problems with various coupled constraints is adopted to optimise the layout parameters. The optimal results can be

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Fig. 5. Layout optimization process for AFP mechanisms.

obtained from the method, if convergence is reached. If convergence does not exist, one must modify the corresponding established models and algorithms until the optimal variables is obtained. 4.2. Mathematical modelling for AFP layout optimisation According to the analysis of Section 2, the AFP module can be divided into three subsystems, and the objectives and constraints are provided for each subsystem. Furthermore, the constraints and global objective function between the adjacent subsystems are given in the next subsections. 4.2.1. Upper bearing plate On the upper bearing plate, the creel assembly is uniformly distributed, which is helpful to reducing the inertia of the system. Considering that n bundles are delivered to mould the part, the number of layout objects on each bearing plate is n. In addition, all the objects should be contained within the AFP system, with no overlapping of the upper bearing plate and no clashing between the fibre channel and each subsystem.

⎧ F ind : Xi = {xi , yi } ⎪ ⎪ n

 ⎪ ⎪ ⎨Min : f1 (xi , yi ) = max x2i + y2i i=1

s.t. : Xi − Xi−1  ≥ 2r ⎪ ⎪  ⎪ ⎪ ⎩Xi  2≥π ri + r α= n

(5)

where r is the maximum radius of the creel assembly, n is the number of creel assemblies and r is the radius of the fibre channel. (xi , yi ) is the centre of the creel assembly in coordinate system X O Y . Once the centre of the specific creel assembly is determined, the rest of creel assemblies can be calculated using this geometric relationship. The objective functions denote that the objects are located within a minimum space. 4.2.2. Middle bearing plate In the middle bearing plate, all the tension measurement units should be contained within the plate, with no overlapping of the middle bearing plate and no clashing between the fibre channel and modules. The local objective function here

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optimises the occupied space of the layout objects.

⎧ n F ind : X2 = ∪ {mi , ni , θi } ⎪ ⎪ i=1 ⎪ ⎪ n  ⎪ ⎪ ⎨Min : f2 (mi , ni ) = m2i + n2i i =1 5   5   T

T T

T ⎪ ⎪ s.t.min ∪ max Pjl − Pik nik , M j − Pik nik |k ∈ IBi , ∪ max Pik − Pjl n jl , Mi − Pjl n jl |l ∈ IB j ≥ 0, i = j, i, j ∈ Ig ⎪ ⎪ k=1  5 l=1   5  ⎪    ⎪ T ⎩min ∪ max C − P n |k ∈ I , ∪ max (P − C )T k  |l ∈ I ≥ 0, i = j, i, j ∈ I l

l=1

jk

ik

BC

k=1

ik

l

g

Bj

jl

(6)

where IB j =IBi ={1,2,3,4,5} and IBc ={1,2,3,4,5,6} refer to the vertices of layout objects and fibre channel, respectively. (mi , ni )

is the centre of the ith object in coordinate system X O Y , and θ i is the directional angle of the ith tension measurement unit, which is defined in Fig. 4. 4.2.3. Lower bearing plate In the lower bearing plate, CCR units must be contained within the module with no overlapping among the subsystems and no clashing between the fibre channel and each subsystem. The objective function can be expressed as follows:

⎧ n F ind : X3 = ∪ {wi , ei , γi } ⎪ ⎪ i=1 ⎪ ⎪ n  ⎪ ⎪ ⎨Min : f3 (wi , ei ) = w2i + e2i i =1 5   5   T T T T ⎪ ⎪ s.t.min ∪ max Q jl − Qik tik , W j − Qik tik |k ∈ IBi , ∪ max Qik − Q jl t jl , Wi − Q jl t jl |l ∈ IB j ≥ 0, i = j, i, j ∈ Ir ⎪ ⎪ k=1  5 l=1   ⎪    ⎪ 5 ⎩min ∪ max (C − Q )T t |k ∈ I , ∪ max (Q − C )T k |l ∈ I ≥ 0, i = j, i, j ∈ I l

l=1

ik

ik

Bi

ik

k=1

l

l

BC

(7)

r

where (wi , ei ) is the centre of the specific CCR unit in coordinate system XOY, and γ i is the directional angle of the specific CCR unit defined in Fig. 4. 4.2.4. Common constraints analysis between the adjacent bearing plates As mentioned in Section 2, the fibre tension should be precisely controlled during the laying process. If the fibre transmission distance is shorter, the servo motor can more effectively control the fibre tension, reduce the potential for defects and increase the layup reliability [47]. To efficiently control the fibre tension, one must determine the transmission distances for different possible layouts. Hence, the first objective function is formulated so as to calculate the fibre transmission distance for a specified layout, as shown in Eq. (8).

L=

n      X i − M i  + M i − W i 

(8)

i=1

In consideration of the compressive strength of the prepreg, the fibre tow must be transported more smoothly in order to reduce the gap or overlap in the fibre production. As tows are threaded through the above mechanism, tows will inevitably bend and twist several times and slide on the guide roller, which may lead to physical damage. Further, fibre tows will finally be gathered as a band, assuming that it is parallel to plane-ZOY. To simplify the objective, the bending angle of a fibre tow can be approximately represented by using the output point of each object and the objective function is formulated as follows:

f (εi ) = ε1 +(ε2 +ε3 )+|ε4 | where arccos(ε1 ) = −−→ −−−→ Xi Xi and Xi  Mi  .

(9)

−−−−→ −−→ −−−−→ M  W  ·(X Xi  ×Xi  Mi  ) arcsin( −−−i−→i i −− → −−−−→ ) ||M W  ||×||X X  ×X  M  || i

i

i i

i

−−−−→ represents the angle between Mi Wi  and the plane that is composed of

i

−−−−→ −−→ −−−→ −−→ Mi Mi  ·Xi Xi  −−−−→ −−→ ) is the twisting angle between Mi Mi and Xi Xi ,   ||Mi Mi ||×||Xi Xi || −−−−→ −−−→ −−−→ −−−→ Mi Mi  ·WiWi  arccos( −−−−→ −−−→ ) is the twisting angle between Mi Mi and WiWi .   ||Mi Mi ||×||WiWi || −−−−→ −−→ Q Q ·OC6 arccos( −−−i1−→i5 −−→ ) is the angle between the X-axis and the final fibre Qi1 Qi5 OC6 

arccos(ε2 ) = arccos( arccos(ε3 ) = arccos(ε4 ) =

orientation on the S3 bearing plate.

Here, the design objective is to optimise the range of the fibre bending degree (the smaller the better) while satisfying the following constraints between the two plates. First, the transmission routes of each fibre tow should not interfere with each other, which means that no values for t1 , t2 , t3 , and t4 satisfy the following:

 ⎧    xi + mi − xi t1 = xj + mj − xj t2 ⎪ ⎪



 ⎨y + n − y t = y + n − y t j i i 1 j 2 

i 

j  + m − w t = w + m − w t ( 0 ≤ ti ≤ 1 , j = i ) w ⎪ 3 i i i j j  i 4 ⎪  ⎩       ei + ni − ei t3 = e j + n j − wi t4

(10)

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Fig. 6. The adopted multidisciplinary design optimization: nested analysis and design (NAND).

where 0 X  i = (xi , yi , z2 )T and 0 X  j = (x j , y j , z2 )T are the output points of the ith and jth creel assemblies of the upper bearing plate in the coordinate system Oxyz , respectively; 0 M  i = (mi , ni , z1 )T and 0 M  j = (m j , n j , z1 )T are the output points of the ith and jth tension measurement units of the middle bearing plate, respectively; 0W  i = (wi  , ei  , z0 )T and 0W  j = (w j  , e j  , z0 )T are the output points of the ith and jth CCR units of the lower bearing plate, respectively. The approaches of MDO design problems can be summarised as single-level optimization and collaborative optimization [43–46]. We select the nested analysis and design approach (a kind of single-level approach) [44,46], shown in Fig. 6. Each disciplinary objective is executed in parallel. The optimization problem is solved in the all-at-once approach as a nested execution. Hence, on the basis of the above process, through the computation of weighting and dimensionless, the layout optimization objective of the AFP is transformed into a single objective function by giving the weight coefficients expressed as:

F (xi , yi , mi , ni , θi , wi , ei , γi ) =

f1

σ1

n x2max + y2max

+ σ3 

 + σ2   n

f3



n w2max + e2max

f2 m2max

 + σ4

+ n2max



L f ( εi ) +σ5 max L max f (εi )

(11)

where the weight coefficients σ 1 , σ 2 , σ 3 , σ 4 , σ 5 reflect the importance of each optimization objective in the AFP mechanism, thus these coefficients are determined by the designer’ focus and functional requirements. Therefore, the layout optimization model is established as follows:

⎧ n ⎨Find : X = ∪ {xi , yi , mi , ni , θi , wi , ei , γi ...} i=1 ⎩Min : F (xi , yi , mi , ni , θi , wi , ei , γi ... ) s.t. fi (X ) ≥ 0i = 1, 2,...,n

(12)

5. Simulation results and discussion 5.1. Experiment settings We provide two examples of AFP mechanisms to verify the proposed methodology. Example 1 is an AFP mechanism with 4 bundles of fibres, including 12 objects (4 cylinders and 8 convex polyhedrons) attached to the three bearing plates S1 –S3 . Example 2 is an AFP mechanism with 6 bundles of fibres and contain more subsystems (6 cylinders and 12 convex polyhedrons). These two cases are studied for illustrating the implementing process. This study focus on optimizing the fibre transmission path and mitigating fibre damage, which are equally important, so that the weight coefficients of each objective in two cases are assigned as 1, formulated as,

F (xi , yi , mi , ni , θi , wi , ei , γi ) =

f1

n x2max + y2max +

 

+  n

f3

n w2max + e2max

f2 m2max

+

+ n2max



L f ( εi ) + max L max f (εi )

(13)

The differential evolution (DE) algorithm has the outstanding abilities [33,34] in solving engineering optimization, so that we adopt DE algorithm to search optimization layout results. The comparison with standard layout design and the proposed

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optimization results are given with specific analyses and discussions. In the following the layout optimization process is taken as example in detail. Step 1: Initialise populations: the first generation {xi (0)| xL ji ≤xi (0) ≤ xU ji , i = 1, 2,…, NP ; j = 1, 2, 3,…, D} is randomly generated. NP and D represents the population size and the number of design variables, respectively. Step 2: Mutation operation: VG+ 1 = XG (p) + F∗ (XG (j) − XG (k)) where G is the Gth generation during the optimization process and F denotes the mutation possibility. V (i, j ) i f rand(0, 1 ) ≤ Cr or i=irand Step 3: Crossover operation: UG+1 (i, j ) = { G+1 irand denotes a random integer of [1, 2, XG (i, j ) else 3… D]. Cr is the crossover probability. U (i ) i f ϕ (UG+1 ) ϕ (XG ) Step 4: Selection operation: XG+1 (i ) = { G+1 XG+1 (i) is the ith population of offspring XG+1 ; ϕ is XG (i ) else the fitness function of optimization. This procedure will continue to run until the convergence of layout optimization is reached. 5.2. Optimization results and discussion Example 1. The parameters for DE in this case are as follows: Np (population size) is 250, G (iteration times) is 90 0 0, F0 (mutation probability) is 0.2, and Cr (crossover probability) is 0.8. Keep in mind that all the DE parameters are provided through a broad range of trials and experience [48], and the stopping criterion is given as (ϕ G− G −ϕ G )/ ϕ G − G ≤ ε or G = Gmax , where ϕ G is the average value of the fitness function in the G generation population, ε is set as 0.01, and G is set to 20 0 0. All the computation and experiments are implemented by MATLAB R2017b. The constant parameters of each object on the three plates are shown in Table 1. The optimization on the Z-axis for the AFP mechanism is not significant in this study. The main objective is to minimise the bending degree, length and the occupied space within the AFP mechanism in such a way that the layout objects have no overlap. The differential evolution algorithm is then adopted to optimise the variables of the established model. The initial and final iteration groups of the optimal values are presented in Table 2. A typical traditional layout design is also given for comparison and the corresponding parameters are listed as follows. In Fig. 7, the results of the initial layout scheme and optimised layout scheme in each bearing plate is presented by dotted line, and the green, blue, black and cyan colours represent the objects of the different transmission paths. The orientation and position of each subsystem have been optimised in different bearing plates and the optimised fibre properties of the AFP mechanism are shown in Table 3. Since fibre tows will finally be gathered as a band that is parallel to the ZOY plane, the layout objects are well-distributed and gradually parallel along the X-axis. The relationship of the iteration and the optimal function values F are given in Fig. 8. With increase of the iterative generation, the objective results that are monotonically

Table 1 Constant parameters of the module. Parameters Length(mm)

z0 7.5

z1 67

z2 197

Pi1 Pi2 

Pi2 Pi3 

Pi3 Pi4 

Pi4 Pi5 

Pi5 Pi1 

18

75

25

75

18

Parameters

r0

r1

Qi1 Qi2 

Qi2 Qi3 

Qi3 Qi4 

Qi4 Qi5 

Qi5 Qi1 

Length(mm)

48

57

26

176

36

196

20

Table 2 Results of the optimal layout design. Iterative generation

Iterative generation

Parameters (time)

k = 271

k = 3150

Traditional layout

Parameters (time)

k = 271

k = 3150

Traditional layout

xi (mm) yi (mm) m1 (mm) n1 (mm) θ 1 (rad) m2 (mm) n2 (mm) θ 2 (rad) m3 (mm) n3 (mm) θ 3 (rad) m4 (mm) n4 (mm)

75.9 84.4 46.7 101.2 4.0 −96.2 71.1 5.6 −51.5 −73.8 5.6 105.1 −52.2

82.6 72.7 68.9 68.3 4.2 −63.2 71.2 5.1 −73.7 −64.5 1.0 62.7 −71.1

78 78 2.5 93 −1.57 17.5 −93 1.57 −15.5 −93 1.57 −31 93

θ 4 (rad)

2.3 109.1 61.0 5.07 −114.1 115.1 5.0 −99.6 −65.4 6.0 111.0 −103.0 0.3

71.4 76.9 57.8 61.5 3.9 −52.1 61 5.6 −60.5 −62.6 0.8 71.1 −54

−1.57 13 151 −1.57 34 −151 1.57 −10 −151 1.57 −31 151 −1.57

w1 (mm) e1 (mm) γ 1 (rad) w2 (mm) e2 (mm) γ 2 (rad) w3 (mm) e3 (mm) γ 3 (rad) w4 (mm) e4 (mm) γ 4 (rad)

10

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Fig. 7. Initial layout schemes and optimised layout schemes in the 4-strip AFP mechanism model. Table 3 The fibre properties of the AFP mechanism corresponding to the best value. Transmission length of fibres (mm) Fibre bending degree

703.93

ε1

ε 2 +ε 3

|ε 4 |

4.0

7.5

3.9

Overlapping A(mm2 )

0

decreasing are represented by a blue coloured line, and the results are relative stable at −1.673 after k = 3600 iterative generations. Therefore, the optimised model reached a global optimal result. After being optimised, the size and shape of fibre channel will be slightly adjusted, and other auxiliary modules can be arranged on the basis of the optimal layout design. Further the optimised results listed in Table 2 are improved by 52.9%, when compared to the results of traditional layout −1.092. Example 2. The parameters for DE in this case are as follows: Np is 380, G is 30 0 0, F0 is 0.02, and Cr is 0.8, and the stopping criterion is the same as the previous one. It can enhance the necessity and importance of the AFP mechanism layout optimization and help to verify the DE algorithm efficiency. The 6-strip AFP layout results have been obtained, as shown in Table 3, and the corresponding 2D diagram are listed in Fig. 9(b). For comparison, a typical traditional layout is also listed in Table 3 and Fig 9(c). The iteration curve of the

W. Zhang, F. Liu and Y. Lv et al. / Mechanism and Machine Theory 144 (2020) 103651

Fig. 8. Number of iterations versus optimal function values.

Fig. 9. Initial layout schemes and optimised layout schemes in the 6-strip AFP mechanism model.

11

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Fig. 10. Number of iterations versus optimal function values.

Table 4 Results of the optimal layout design. Iterative generation

Iterative generation

Parameters (time)

k = 13

k = 1070

Traditional layout

Parameters (time)

k = 13

k = 1070

Traditional layout

xi (mm) yi (mm) m1 (mm) n1 (mm) θ 1 (rad) m2 (mm) n2 (mm) θ 2 (rad) m3 (mm) n3 (mm) θ 3 (rad) m4 (mm) n4 (mm) θ 4 (rad) m5 (mm) n5 (mm) θ 5 (rad) m6 (mm) n6 (mm)

96.0 96.0 −147.7 −48.4 4.7 −76.2 8.6 3.8 22.0 79.8 1.5 98.9 12.7 4.6 137.3 −75.6 4.1 −9.0 158.0

93.9 81.0 63.8 74.8 4.1 −19.3 97.5 4.8 −87.9 47.9 5.6 −66.6 −72.6 1.0 15.4 −96.8 1.7 95.6 −47.6

62 108 −38 111 −1.57 −8 111 −1.57 22 111 −1.57 −22 −111 1.57 8 −111 1.57 38 −111

θ 6 (rad)

3.4 129.6 −104.1 2.8 −36.4 −169.1 5.7 −35.1 119.9 5.5 −110.7 −63.4 2.8 97.6 57.0 1.1 101.5 −160.9 3.5

2.4 64.3 127.1 4.4 −21.9 145.9 4.8 −94.3 103.0 5.2 −69.4 −124.2 1.3 15.4 −145.1 1.6 107.2 −101.8 2.1

1.57 −63 163 −1.57 −24 163 −1.57 15 163 −1.57 −31 −163 1.57 8 −163 1.57 47 −163 1.57

w1 (mm) e1 (mm) γ 1 (rad) w2 (mm) e2 (mm) γ 2 (rad) w3 (mm) e3 (mm) γ 3 (rad) w4 (mm) e4 (mm) γ 4 (rad) w5 (mm) e5 (mm) γ 5 (rad) w6 (mm) e6 (mm) γ 6 (rad)

Table 5 The fibre properties of the AFP mechanism corresponding to the best value. Transmission length of fibre (mm)

1051.0

Fibre bending degree

ε1

ε 2 +ε 3

|ε 4 |

5.9

11.5

5.6

Overlapping A(mm2 )

0

optimization process is as shown in Fig. 10. The fibre properties of the AFP mechanism corresponding to the best value are given in Table 5. The optimised results listed in Table 4 show that the best improvement of fibre path obtained by the proposed DE algorithm (the best value of fitness 2.26) is improved by 31.5%, when compared to the traditional layout 3.29. It can be seen from the 2D diagram of the best layout results obtained by DE algorithm that (a) the layout components are symmetrically assigned within the minimal space; (b) these components are gradually parallel to the x-axis from S1 to S3 . The above characteristics of layout are consistent with the expectation based on the presupposed conditions and features. From the above analysis, it can be seen that the proposed assignment and the layout optimization strategy have to be improved in the optimization process. Future work will emphasise on the study of improvements of the AFP mechanism’s layout optimal algorithms and make a comprehensive comparison and discussion on success ratio, computational time and effectiveness, so as to entirely search the better solution space of objective functions. Considering actual properties of fibres and subsystems, the multiobjective optimization mechanism is required for the coordination with the objects assignment and layout optimization algorithm. The experiments for the comparison of two configurations in the AFP mechanisms will be carried out in the next step.

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13

6. Conclusion This paper presents the decomposition of the AFP mechanism and the layout design optimization method for reducing fibre defects while following the fibre transmission path. The minimal length of the fibre path and bending deformation within the AFP mechanism are important impacts and are defined as optimization objectives for reducing fibre defects. The entire AFP mechanism model is first established and analysed. To accomplish single fibre tow control, logical relationships between each module are taken into account; then, the mechanism is decomposed into four parts: the creel assembly unit, the tension measurement unit, the CCR unit and the auxiliary subsystem. The coordinate transformation that is introduced herein represents the orientation and position of the layout modules within the AFP mechanism on the basis of system analysis. A global optimization with nested analysis and design approach is developed and successfully applied for the AFP mechanism layout. Due to the characteristics of the multi-constraints and the strong coupling of the AFP mechanism, the objective is to minimise the occupied space and reduce fibre defects and bending while following the transmission path. The constraints are established to prevent interference from occurring between the adjacent objects and fibre path, and these constraints balance the requirements in a global system. Based on the above analyses, two cases for the AFP mechanism is given. The layout results show that the global objective and degree of fibre bending have been optimised. The coordinate transformation is proven to be an efficient technique for stating and simplifying the layout design problem, and the above optimization process provides a novel method to guide the layout design for such a complex system. Declaration of Competing Interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. 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