Randomized multi-objective optimal design of a novel ... - CiteSeerX

Original Article

Randomized multi-objective optimal design of a novel deployable truss

Proc IMechE Part G: J Aerospace Engineering 227(11) 1720–1736 ! IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954410012461874 uk.sagepub.com/jaero

Hailin Huang1,2, Bing Li1,2, Zongquan Deng3 and Rongqiang Liu3

Abstract In this article, a novel deployable mechanism that can be depaloyed from a bundle compact configuration onto a large volume double-layer truss structure is proposed. The mechanism is constructed by a set of Myard linkages through specially designed mechanical connections, so that the whole assembled mechanism has single degree of freedom. The model of the multi-objective design for the proposed deployable mechanism is developed. In the optimal design of this mechanism, many design objectives have to be taken into consideration, such as weight, stiffness, packaging/expansion ratio and natural frequency, etc. Many of these design objectives have no explicit analytical expression and may be contradicted with each other. A randomized multi-objective search algorithm is proposed for solving this multi-objective design problem, by using the algorithm, the set of Pareto optimal solutions can be obtained, and the relationship between different objectives is figured out, so that the designers can choose the compromise solutions intuitively. The physical prototype is also fabricated based on the optimized parameters, the stiffness and natural frequency experiments are conducted to evaluate the design. The experimental results demonstrate that the proposed mechanism offers an attractive combination of performance characteristics for both stiffness and natural frequency. Keywords Large deployable mechanisms, multi-objective design, swarm intelligence, Pareto optimal solutions Date received: 1 February 2012; accepted: 29 August 2012

Introduction Deployable mechanisms are widely used in both aerospace applications and general civil engineering applications.1 In the design of deployable mechanisms, the planar mechanisms such as scissor shape mechanisms or parallelogram mechanisms are often selected as the basic building element for the construction of large deployable truss mechanisms: Zhao et al.2 used scissor shape mechanisms for constructing foldable stairs; Kiper et al.3 used the planar deployable units to form a class of deployable polygons and polyhedra; Katherine4 used the Hoberman mechanism to design the reconfigurable antenna and solar arrays; Ozawa et al.5 used the planar linkage to construct 30 m deployable signal reflector in the engineering test satellite; Gantes,6 Hanaor and Levy7 and Pellegrino8 have also done tremendous work on the deployable structures for civil engineering applications. Recently, many researchers have begun to pay special attentions to the new type of spatial mechanisms

for the design of deployable mechanisms. Lin et al.9 proposed a new deployable and lockable mechanism for planetary probes; Chen10 developed a class of deployable mechanisms based on the Bennett linkage, Myard linkage and the Bricard linkage; Luo et al.11 used the Bricard linkage for retractable structures in civil engineering; Baker12 presented a class of foldable spatial 6 R mechanisms based on the Bricard linkage; the authors of this article have given a systematic 1 Shenzhen Graduated School, Harbin Institute of Technology, Shenzhen, People’s Republic of China 2 Shenzhen Key Laboratory of Advanced Manufacturing Technology, Shenzhen, People’s Republic of China 3 School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, People’s Republic of China

Corresponding author: Bing Li, Shenzhen Graduated School, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of China. Email: [email protected]

Huang et al. synthesis for the spatial deployable single-loop mechanisms based on Lie group theory,13 and the mobile assembly of single-loop deployable mechanisms are also synthesized systematically;14 Cui et al.15 used the general line-symmetric Bricard linkage for the construction of large modular surface deployable antenna structure; Gioia et al.16 developed the foldable/unfoldable corrugated architectural curved envelop based on the spherical linkages. The spatial single-loop mechanisms are ideal candidates for aerospace applications, as they can provide good stiffness, low packaging/expansion ratio and easily be extended to large scale deployable networks. In this article, a novel double-layer deployable mechanism that can be deployed from a bundle compact configuration onto a large volume truss structure is presented. The mechanism is constructed by a set of Myard linkages through special mechanical mobile assemblies, so that the whole assembled mechanism has single degree of freedom (DOF). Different from the general industrial robotic mechanisms, the design of this deployable mechanism is complicated in that it consists of a large set of basic deployable units, all of which could be designed with different geometric parameters, such that the design parametric space is very large. In the design process it simultaneously involves the multiple design objectives such as weight, stiffness, packaging/expansion ratio and natural frequency, etc., designers always want to know the relationship between the multiple design objectives, so that they can select a compromise solution between the contradicting objectives; however, many objectives do not have explicit analytical expressions. Solving for Pareto optimal set for the multi-objective design (MOD) problem is a time consuming process as the dimension of the parametric space is very large, and the evaluations of the objective functions are complicated due to the presence of a large number of sub-optimal Pareto solutions. It is well-known that the swarm intelligence algorithms are efficient methods for dealing with the optimization problems with large number of sub-optimal.17 Inspired by the idea of swarm intelligence optimization, a high efficient randomized swarm intelligence search algorithm is proposed in this article. In the proposed algorithm, the modified controlled random search (CRS) algorithm18 is used as the basic searching individuals, the nondominance is used as its evaluation criterion, and the objectives are factorized into several stages to get the compromise solutions during the evaluation. Using this algorithm, the interrelationship between all design objectives and parameters can be figured out, which provides an intuitive way for designers to choose the compromise solutions in the solution space. The rest of this article is organized as follows: in the following section, the novel double-layer deployable

1721 mechanism is proposed; in the third section, the MOD problem of large deployable mechanisms is formulated; in the fourth section, the randomized multiobjective searching algorithm is developed; in the fifth section, the MOD problem is tested under the proposed algorithm; in the sixth section, physical prototype is shown, and the stiffness and natural frequency experiments are conducted; the conclusions to this article are given in the last section.

Proposal of the double-layer deployable mechanism The physical model of deployable Myard unit and the corresponding parameters are shown in Figure 1(a), it consists of five revolute joints A, B, C, D, E and five links. The Denavit-Hartenberg (D-H) model of this mechanism is as shown in Figure 1(b), a, b, c, d are the 4 D-H links for the mechanism. a, d are called short links of the mechanisms, while b, c are called long links of the mechanism; Z1 , Z2 , Z3 , Z4 , Z5 are the axes of revolute joints A, B, C, D, E, respectively; the origin o  xyz is set with z-axis being perpendicular to Z3 , Z4 and x-axis being along the bisector of angle hZ4 , Z3 i; 1 , . . . , 5 are the five joint variables of the mechanism; the mechanism is mirror-symmetric about the xz plane, each link of physical model has identical length with the corresponding link in the D-H model. In this article, we only focus on the Myard unit with its twist angles satisfying equation (1) 8 < 34 ¼ hz3 , z4 i ¼ 2=3  ¼ hz4 , z1 i ¼ 23 ¼ hz2 , z3 i ¼ =2 : 41 15 ¼ hz1 , z5 i ¼ 52 ¼ hz5 , z2 i ¼ =6

ð1Þ

Based on Myard,19 the closure equation of the Myard 5R mechanism is given as 8 2 ¼ 1 , 3 ¼ 4 þ > > > > > 1  34  > > þ sin > >  > 2 2 2 tan 4 < tan 1 ¼   1  34  2 2  sin > > 2 2 2 > > > > > b ¼ c, a ¼ d > > > : a ¼ b cos 34 =2

ð2Þ

Chen and You20 has presented an umbrella shape deployable mechanism constructed by Myard mechanism, and a larger single-layer deployable network is constructed by such kind of umbrella shape deployable structure,21 in which every three umbrella modules

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Proc IMechE Part G: J Aerospace Engineering 227(11)

(a)

A

Z1

B

d

a

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θ1

Z1

Z2

N

M-M section R1

M M b

Z6 z

b

r1

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Z4

P-P section h

c

N-N section R2

P

Z4

d θ5

a

E

N

Z2 θ2

Z3

θ4

x

f

Figure 1. Geometric parameters of the deployable Myard mechanism: (a) physical model; (b) D-H model.

u1

B2

A1

Upper 3-Myard mechanism

A2

u2

O

A3 B ' 2 u3 B3

B1

u2'

u1' B1'

A2'

A 1' O'

B3' Lower 3-Myard mechanism A3' u3'

Figure 2. Conceptual model for one module of the double-layer deployable structure.

connected by three additional revolute joints are concurrent at a point. However, such deployable network cannot achieve good stiffness due to the single layer structure. In this article, we will construct a novel double-layer deployable truss structure by using the Myard mechanism units. Figure 2 shows one module of the proposed double-layer deployable truss, every three identical Myard units are connected by three additional revolute joints A1 , A2 , A3 symmetrically, so that their axes u1 , u2 , u3 can concurrent at a point O, after this connection, the 3-Myard mechanism still has one DOF, because this mechanical structure are equivalent

to the threefold-symmetric Bricard mechanism.22 By using another identical 3-Myard mechanism (lower 3-Myard mechanism in Figure 2), which is set to be the images of the upper 3-Myard mechanism, one can see 0 0 0 that the three bases B1 , B2 , B3 of the lower mechanism have identical motion trajectory with the three bases B1 , B2 , B3 of the upper mechanism. In other words, 0 one can combine Bi into Bi ði ¼ 1, 2, 3Þ as one rigid body without changing the mobility of the respective two 3-Myard mechanisms, as long as the image-symmetric is retained for the lower and the upper 3-Myard mechanisms. As has been mentioned in Huang et al.,4 the aforementioned 3-Myard mechanism can be used as the basic module to construct larger modular deployable network, so that the deployable mechanism can be infinitely be extended. The mechanism can be designed as a modular mechanism with single DOF For example, we can array six modules of such mechanism symmetrically to form a larger deployable network, which can be deployed from the bundle compact configuration (Figure 3(a)) into the large volume deployed configuration with double-layer truss structure (Figure 3(b)).

Formulation of the MOD model for large deployable mechanisms In the engineering practice of large deployable mechanisms, the main property requirement is the structural dynamics, particularly in the deployed configuration. Generally, the MOD model can be established based on the following aspects: minimized mass W of the whole mechanical system; minimized packaging/

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Figure 3. Deployable mechanism with six modules: (a) folded configuration; (b) deployed configuration.

expansion ratio ; maximized the stiffness K and natural frequency ! to avoid the resonance response; some other static constraints, such as distorting stress  and bending stress , etc. can be included in the model. The MOD problems can be formulated as finding the suitable solution x in the design parametric space X, the model is given as follows min

WðXÞ, ðXÞ, 1=KðXÞ, 1=!ðXÞ

s:t: ji j4½ , i ¼ 1, 2, . . . , n   j 4½, j ¼ 1, 2, . . . , m

ð3Þ

min xmax k 4xk 4xk , xk 2 X

However, some of the design objectives do not have explicit analytical expressions and many of them may be contradicted with each other, the solutions for the optimization problem are a solution set called Pareto optimal set, the optima values of some objective functions may yield poor values for other objectives. In this article, the multi-objective optimization problem is transferred into the problem of finding interrelationship between the multiple objectives and design parameters, which may provide the designers an intuitive way to choose the parameters based on the specific requirements. In this article, the following design objectives are considered.

launch vehicle; it also has an important effect on the dynamic properties, such as the active forces of the system. For the aerospace applications where the gravity is nearly negligible, the work of the system mainly comes from overcoming the inertia and the damping of the mechanical system. Thus, the mass of the mechanism is expected to be reduced. For the deployable mechanism studied in this article, the total mass of the 6-module deployable network can be calculated by the summation of all long links ML , short links MS and bases MB , which can be respectively calculated by 8   M ¼ l  R21  r21 b > > < L   MS ¼ l  R22  r22 a > > pffiffiffi : MB ¼ B 3 3f 2 h=2

ð4Þ

There are totally 7 bases, 12 long links and 24 short links in the center module, 36 long links and 48 short links are composed in the peripheral modules, it is supposed that the radial parameters of the links for center module are different from the peripheral modules, so that the total mass can be calculated as       W ¼ 12  R21  r21 b þ 24  R22  r22 a   0   0 0  0  þ 36  R12  r12 b þ 48  R22  r22 a pffiffiffi 3 3f 2 h þ 7B 2

ð5Þ

Total mass of the mechanism

Packaging/expansion ratio

The total mass of the deployable mechanism is an important consideration due to the limitation of the

Due to the launch vehicle limitation the packaging/ expansion ratio is considered as an important objective

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(a)

q3 q1q 4 q6 2 q2 q5

(b) q16 q 21

q7

q12 q9 3 q13 q11 q10 q q14 8

q19

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9 16 10 19

2 1 7

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15 18

20 22

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17

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X

Figure 4. Finite element model of the mechanism: (a) coordinates of single Myard linkage; (b) elements for one module of the truss mechanism.

in the design of the deployable mechanism. The packaging/expansion ratio can be calculated as the envelope volume of the mechanism in the folded configuration divided by the envelope volume of the mechanism in the deployed configuration. As shown in Figure 3(a), the envelope volume for deployed configuration is the minimum column that containing the deployed mechanism. Similarly, the envelope volume for the folded configuration is the minimum column that containing the folded mechanism as shown in Figure 3(b). Then the packaging/expansion ratio is given as  pffiffi 2  4uþ32 3f ð2b þ hÞ ¼  pffiffi 2 3f  4b cos ’þ3 ð2b sin ’ þ hÞ 2

ð6Þ

where ’ is the half angle of each long link with respect to its corresponding mirror-image long link in the deployed configuration.

Stiffness, natural frequency and loading capability The stiffness measures the resistance to the deformation of the deployable mechanism under the loads. For the aerospace applications, the deployable mechanisms are often designed with large volume; the elastic deformation must be taken into account. The natural frequency of the mechanism is expected to be maximized to avoid the resonance response. The stress for the each link of the mechanism should also be reduced to avoid the possible damage of the components. The natural frequency, stiffness and the stress can be solved from the finite element model. The dynamic equation

containing the elastic deformations and the damping components can be given as Mq€ þ Cq_ þ Kq ¼ F  Mq€ r

ð7Þ

Where M is the mass matrix, K is the stiffness matrix, q€ r is the rigid acceleration vector, q is the elastic deformation vector, C is the damping matrix and can be given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 C ¼ MC0 , Cii ¼ 2i Kii =Mii , Cij ¼ 0, i 6¼ j ð8Þ As the large deployable network is a multi-body mechanical system, the finite element method can be employed for the structural analysis. As shown in Figure 4(a), one Myard mechanism contains 4 link elements and 22 coordinates q1  q22 ; it supposes that the base is a rigid structure without any elastic deformation. Then the coordinates for one representative submechanism is shown in Figure 4(b), as the mechanism is line symmetrical around Z-axis, the whole mechanism can be set similarly by arranging six sub-mechanisms around Z-axis. In the deploying process, the coordinates in every link for the sub-mechanism can be set as given in equation (9), the first column out of the bracket represents the number of the link elements, the elements of ith row in the bracket is the number of coordinates for the link i. Iu is the transformation matrix that used to reflect the local coordinates into global coordinate. For example, the first row of equation (9) contains 12 elements, the first six elements are zero represent that one end of link 1 is connected to the base rigidly, the seventh to the twelfth elements are 1 to 6, it represents that the coordinates q1  q6 locate at the other end of link 1. By using the typical finite element method, one can easily calculate the coefficient matrices M, C and K.

Huang et al.

1 2 3 4 5 6 7 8 9 10 11 12 Iu ¼ 13 14 15 16 17 18 19 20 21 22 23 24

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

1725

0 1 8 15 0 24 31 38 46 52 59 1 46 38 75 82 92 15 100 52 92 82 115 24

0 2 9 16 0 25 32 39 47 53 60 2 47 39 76 83 93 16 101 53 93 83 116 25

0 3 10 17 0 26 33 40 48 54 61 3 70 40 77 84 94 17 102 54 110 84 117 26

0 4 11 18 0 27 34 41 49 55 62 4 71 41 78 85 95 18 103 55 111 85 118 27

0 5 12 19 23 28 35 42 50 56 63 66 72 73 79 86 96 98 104 107 112 113 119 122

0 7 14 21 0 30 37 44 51 58 65 6 51 74 81 88 97 99 106 57 97 114 121 29

1 8 15 0 24 31 38 0 52 59 1 46 38 75 82 46 15 100 52 92 82 115 24 92

2 9 16 0 25 32 39 0 53 60 2 47 39 76 83 47 16 101 53 93 83 116 25 93

The coordinates given in equation (9) are based on the assumption that one actuator is located at the revolute joint that connecting the base B0 and the link element 1, the mechanism lies on the deploying or folding stage. As soon as the mechanism moves to the deployed

2 0 1 2 6 6 1 8 3 6 4 6 6 15 5 6 6 0 6 6 6 22 7 6 6 29 8 6 36 6 9 6 43 10 6 6 49 11 6 6 56 12 6 61 Iu ¼ 13 6 6 43 14 6 6 36 15 6 6 70 16 6 6 77 17 6 6 85 18 6 6 15 19 6 6 93 20 6 6 49 21 6 6 85 22 6 6 77 23 4 107 24 22

0 2 9 16 0 23 30 37 44 50 57 2 44 37 71 78 86 16 94 50 86 78 108 23

0 3 10 17 0 24 31 38 45 51 58 3 45 38 72 79 87 17 95 51 87 79 109 24

0 4 11 18 0 25 32 39 46 52 59 4 67 39 73 80 88 18 96 52 104 80 110 25

0 5 12 19 0 26 33 40 47 53 60 63 66 68 74 81 89 91 97 100 103 105 111 114

0 7 14 21 0 28 35 42 48 55 62 6 48 69 76 83 90 92 99 54 90 106 113 27

1 8 15 0 22 29 36 0 49 56 1 43 36 70 77 43 15 93 49 85 77 107 22 85

2 9 16 0 23 30 37 0 50 57 2 44 37 71 78 44 16 94 50 86 78 108 23 86

3 10 17 0 26 33 40 0 54 61 3 48 40 77 84 89 17 102 54 94 84 117 26 123

4 11 18 0 27 34 41 0 55 62 4 68 41 78 85 90 18 103 55 108 85 118 27 124

5 12 19 22 28 35 42 45 56 63 66 69 73 79 86 91 98 104 107 109 113 119 122 125

6 13 20 0 29 36 43 0 57 64 67 51 44 80 87 51 21 105 108 97 88 120 123 97

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

ð9Þ

configuration, however, the coordinate setting is different because the mechanism is locked into a solid structure. Suppose that in the deployed configuration, every link connecting to the base is locked rigidly to the base, then the coordinate is given by equation (10) as follows

3 10 17 0 24 31 38 0 51 58 3 45 38 72 79 45 17 95 51 87 79 109 24 87

4 11 18 0 25 32 39 0 52 59 4 65 39 73 80 84 18 96 52 102 80 110 25 116

5 12 19 0 26 33 40 0 53 60 63 66 68 74 81 47 91 97 100 103 105 111 114 89

3 6 13 7 7 20 7 7 0 7 27 7 7 34 7 7 41 7 7 0 7 7 54 7 61 7 7 64 7 7 48 7 7 42 7 7 75 7 7 82 7 7 48 7 7 21 7 7 98 7 7 101 7 7 90 7 7 83 7 7 112 7 7 115 5 90

ð10Þ

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In this article, the stiffness, the loading capacity and the first-order natural frequency at the deployed configuration are considered as the performance evaluation indices.

PgBest L PgBest-Xki

Xik+1=Xki+´Vik+1 Vik+1

Xki

Randomized multi-objective search algorithm

PBesti-Xki

The particle swarm optimization algorithm (PSO) is a typical swarm intelligent search algorithm,23 which can be described as follows: Each particle i represents a possible solution to the optimization task, and each particle is associated with two vectors, the velocity  vector Vi ¼ v1i , v2i , . . . , vD and the position vector i Xi ¼ x1i , x2i , . . . , xD i , where D represents the dimensions of the solution space. During the iteration process, the velocity and the position of the particle i are updated by the following two equations   k k k Vkþ1 ¼ V þ c rand P  X 1 1 i i Besti i   þ c2 rand2 PkgBest  Xki

PBesti G

Figure 5. The iteration of the standard particle swarm optimization algorithm.

Δα1

P' Searching space A

ð11Þ

Δα 2

Δα a

P Δα b B

Xkþ1 i

¼

Xki

þ

Vkþ1 i

ð12Þ

Where rand1 and rand2 are two independent elements from uniform random sequence in the range ð0, 1Þ, c1 and c2 are the acceleration coefficients. PBesti is the position with the best fitness found so far for the ith particle, and PgBest is the position with best fitness of all particles found so far. is the weight coefficient that usually setup to vary linearly from maximum to minimum in due course of iterations. The (k þ 1)-th iteration process for the ith particle can be described by the vector addition operation, as shown in Figure 5. In the search process of PSO, if a particle discovers a local optimal position, all the other particles will move closer to it, this is the basic principle for the regular PSO algorithm. From the basic PSO iteration as shown in Figure 5, one can see that the moving direction of each particle is affected by the position of PgBest . Suppose that there exists one local optima L in the neighborhood of the current PgBest , and the global optima G located in the neighborhood of the PgBesti . If no particle better than PgBest is found, then every particle will move toward the local optima L, the search will converge at the local optima L such that the global optima is ignored. It seems that the moving towards the position PgBest of the current best is not necessary at this case. To overcome this problem, we will use another search algorithm, the controlled

Figure 6. An example of two-dimensional search space.

random search algorithm,24,25 as the basic search particle. In the search process, each particle does not have to compare with the PgBest at each iteration until the local optima in neighborhood of the current position has been found. The basic idea of the controlled random search algorithm can be described as follows ð j Þ ¼ ð j1Þ þ , j ¼ 1, 2, . . . ,

ð13Þ

where  is a vector of the random variables, i , i ¼ 1, . . . , p, it is subjected to normal probability distribution with zero mean and unity standard deviation, i  Nð0, 1Þ, i ¼ 1, . . . , p.  ¼ diagð1 , . . . , p Þ is used to adaptively modify the standard deviation of the normal probability distribution for every random variable in each iteration. If a better objective value has been found, standard deviations are set according to i ¼ K1 i , i ¼ 1, . . . , p. K1 5 1 is used to reduce search interval and maintain search in the neighborhood of the best previous point. Otherwise, after max feval times of no improvement is made with

Huang et al.

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Figure 7. The movements of the particles under PSO algorithm: (a) the initial position; (b) the 82nd iteration; (c) the 181st iteration; (d) the 602nd iteration.

respect to the objective function, the standard deviations are reduced by ði Þnew ¼ K2 ði Þold , i ¼ 1, . . . , p

problem, the value i is set to a positive quantity calculated by

ð14Þ

where K2 5 1 is a positive number. In the traditional CRS, i is a positive quantity describing the distance between the current values of the variables and the nearest bound of the search space. As shown in Figure 6, for example, if the current point locates at P, then there might exist a sequence of inscribed circles that are centering at P and tangential to the boundary of the search space. The circle with minimum radius a will employed as the area for the next search area. However, if the current position lies on the boundary of the search space, for example, P0 in Figure 6, i becomes zero, the next searching area will become the current point P0 itself, such that the algorithm will stop in this point. To overcome this

 ¼

D Y

m =

X

i

ð15Þ

m¼1

where D refers to the dimension of the search space, min m ¼ max m  m is the range of the mth design variable. For example, for the two-dimensional search Q2 space,  represents the envelope search m m¼1 space, the rectangular Q search space with its area calculated by 1  2 , then 2m¼1 m =n represents the average occupied area for each particle. The CRS algorithm has a very high convergence speed to the local optima; however, it is difficult to jump out the local optima because the algorithm itself cannot check the current optima is local or global optima. Inspired by the idea of PSO, if a swarm of

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Figure 8. The movements of the particles under RMOSA algorithm: (a) the initial positions; (b) the 5th iteration; (c) the 10th iteration; (d) the 20th iteration.

    8i 2 f1, 2, . . . , ng : fi Xk 5fi Xk1     9j 2 f1, 2, . . . , ng : fj Xk 4 fj Xk1

ð16Þ

3

x 104 RMOSA PSO

2.5 Number of function evaluations

CRS algorithms are used for the search task, each algorithm has a comparison with others, then the search results can be considered as local optima if any better results are found by other searchers, the individuals locating at local optima are set to jump out the current optima accordingly, but they do not have to move toward the global optima found so far. The iterations in multi-objective optimization are similar to the single-objective optimization, each objective is expected to be improved during each iteration, the nondominance is always used as the evaluation criterion, i.e. in ith iteration, if all objectives f1 , f2 , . . . , fn in the position Xi are no worse than the previous point Xi1 , and at least one objective is better than the previous point, then the previous point Xi1 is dominated by the current point Xi

2

1.5 PSO convergence 1 RMOSA convergence

0.5

0 0

100

200

300 Iteration times

400

500

600

Figure 9. The number of function evaluations with different iteration times under 10 particles. PSO: particle swarm optimization; RMOSA: randomized multiobjective search algorithm.

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Table 1. Pareto optima front of the double-layer deployable mechanism. a(m)

Rc1 (m)

r1c (m)

Rc2 (m)

r2c (m)

Rp1 (m)

r1p (m)

Rp2 (m)

r2p (m)

W(Kg)

!(Hz)

K(N/m)



0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00500 0.00496 0.00473 0.00500 0.00495 0.00500 0.00500 0.00500 0.00500 0.00500

0.00268 0.00377 0.00422 0.00216 0.00405 0.00419 0.00327 0.00401 0.00364 0.00399 0.00406 0.00339 0.00311 0.00363 0.00287 0.00410 0.00201 0.00376 0.00183 0.00435 0.00374

0.00269 0.00200 0.00200 0.00404 0.00203 0.00418 0.00200 0.00206 0.00200 0.00200 0.00470 0.00396 0.00200 0.00304 0.00200 0.00203 0.00201 0.00315 0.00200 0.00200 0.00349

0.00242 0.00050 0.00058 0.00324 0.00176 0.00376 0.00180 0.00118 0.00159 0.00166 0.00423 0.00356 0.00170 0.00273 0.00081 0.00182 0.00144 0.00284 0.00180 0.00172 0.00314

0.00500 0.00472 0.00498 0.00412 0.00461 0.00381 0.00498 0.00472 0.00453 0.00453 0.00366 0.00455 0.00439 0.00441 0.00473 0.00439 0.00433 0.00500 0.00500 0.00388 0.00470

0.00392 0.00425 0.00449 0.00249 0.00415 0.00343 0.00448 0.00425 0.00408 0.00405 0.00329 0.00410 0.00374 0.00397 0.00426 0.00395 0.00294 0.00450 0.00395 0.00349 0.00423

0.00500 0.00376 0.00480 0.00500 0.00447 0.00467 0.00500 0.00442 0.00436 0.00362 0.00464 0.00397 0.00399 0.00420 0.00500 0.00497 0.00500 0.00500 0.00500 0.00429 0.00482

0.00447 0.00339 0.00432 0.00415 0.00402 0.00420 0.00450 0.00398 0.00393 0.00326 0.00418 0.00358 0.00359 0.00378 0.00430 0.00447 0.00431 0.00450 0.00300 0.00386 0.00434

1.1886 0.66472 0.68946 1.5071 0.59897 0.55745 0.79013 0.65075 0.65473 0.56486 0.58115 0.70045 0.77297 0.58583 0.93504 0.58816 1.3455 0.74409 1.6858 0.46231 0.71

57.178 53.114 43.495 39.862 46.119 32.785 48.825 46.549 47.345 55.197 31.234 51.332 55.546 44.791 39.103 38.603 46.178 46.889 34.162 36.245 44.554

7.425E þ 05 3.275E þ 05 4.8E þ 05 8.75E þ 05 4.325E þ 05 4.225E þ 05 6.4E þ 05 4.35E þ 05 4.5E þ 05 2.85E þ 05 4.375E þ 05 3.925E þ 05 4.0E þ 05 3.7E þ 05 7.325E þ 05 4.8E þ 05 8.05E þ 05 6.0E þ 05 10.95E þ 05 3.275E þ 05 5.55E þ 05

0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095 0.095

By using the modified CRS as the local search operator and nondominance as the evaluation criterion, the randomized multi-objective search algorithm is described as follows. Algorithm 1. Randomized algorithm (RMOSA)

multi-objective

search

0: input max_feval, K1 , K2 , number of particle i, ranges of all particles 1: Generate the initial position for each particle i randomly; 2: Evaluate the objectives of each particle under the initial position; 3: repeat 4: for Each particle i do 5: while dominate ðjÞ ð j1Þ 6: Update þ K1 i ;   particle i by i ¼ i ðjÞ 7: if f i dominates f besti then 8: besti :¼ ði j Þ ; 9: dominate ¼ true; 10: else if 11: feval ¼ feval þ 1; 12: if fsd>max_feval 13: ði Þnew ¼ K2 ði Þold ; 14: feval ¼ 0; 15: end if

16: 17: 18: 19: 20: 21: 22: 23:

end if if the stop criteria for this iteration is satisfied then break; end if end while end for until The stop  criterion   is satisfied  output Best besti , f Best besti ;

When each objective is improved slowly or cannot be improved   at all under given times of trials, i.e. fi Xk  fi Xk1  5 ", then the nondominated solution set is approached, this is the stop condition for the algorithm. To show the convergent efficiency of the algorithm, both of the traditional PSO algorithm and the RMOSA with the 10 particles are tested under the test function Griewank. The search processes have been visualized, as shown in Figures 7 and 8. In the figures, the stars represent searching particles, and the global optima locates at the point ð0, 0Þ. We can see that particles move slowly under the traditional PSO algorithm, in more than 600 iterations, the particles under traditional PSO are locating near the global optima, this is because in each iteration, the moving direction of the particles are affected by

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Proc IMechE Part G: J Aerospace Engineering 227(11)

(a)

x 105

7.5

(b) 0.58 0.56 Total mass (Kg)

Stiffness (N/m)

7

6.5

6

0.54 0.52 0.5 0.48

5.5

0.46 5

0

20

40

60 80 Iteration times

100

120

0.44

140

(c) 52

20

40

60 80 100 Iteration times

120

140

(d) 0.102

50

0.101

48 0.100

46 Compactness

Natural frequency (Hz)

0

44 42 40 38

0.099 0.098 0.097 0.096

36 0.095

34 32

0.094

0

20

40

60 80 100 Iteration times

120

140

0

20

40

60

80

100

120

140

Iteration times

Figure 10. Iteration history of the design objectives: (a) iteration history of the stiffness; (b) iteration history of total mass; (c) iteration history of natural frequency; (d) iteration history of compactness.

the one with best fitness found so far. In Figure 7(a), the initial positions of each particle are generated randomly in the searching space, all the objectives are evaluated under the initialized positions, the particle with best fitness is used as the global optimum found so far, then all the particles are moving to this position. In Figure 7(b), the 82nd iteration, one local optima is found to be the best fitness found so far, all the particles move to the new local optima. The algorithm goes on its searching, in the 181st iteration, another local optima that is better that the optima found in the 82nd iteration is found, then all the particles tend to moving toward this position (Figure 7(c)). In the 602nd iteration, the global optima has been found as the best fitness found so far, then all the particles move to the global optima, but there are still some particles far from the global optima. This process is time-consuming. For the RMOSA, each particle aims at finding the local optima as soon as possible in each iteration, this is because the search algorithm is based on the randomized direct search algorithm. In the ith iteration, all

particles will find out the local optima in the neighborhood of the previous best position Xi1 , then all the particles except the best found so far are set to jump out the previous search and to find better fitness. Under this search scheme, we can see that after 5 times of iterations (Figure 8(b)), all the particles are locating at the local optima, with some of it in the global optima, as long as one particle is locating in the global optima, the algorithm is convergent. This fact shows that the RMOSA has great improvement on the computing efficiency by comparing to the traditional PSO. The computation time of the algorithm will be increased with the increasing of iteration times and the number of objective function evaluations. Given the size of the searching particles as 10, we have tested the number of objective function evaluations with respect to different iteration times, as shown in Figure 9. From the figure, we can see that the number of objective function evaluations for PSO is increased linearly with the increasing of the iteration times, this is

Huang et al.

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(a) 7.2

× 105

(b) 60

7 55 Natural frequency (Hz)

6.8 Stiffness (N/m)

6.6 6.4 6.2 6 5.8

55

50

45

5.6 5.4 5.2 0.46

0.48

0.5

0.52

0.54

0.56

0.58

40 0.5

0.6

0.55

0.6

0.65

Total mass (Kg)

(c) 7.2 x 10

5

49 48 Natural frequency (Hz)

7

Stiffness (N/m)

6.6 6.4 6.2 6 5.8

0.9

0.95

47 46 45 44 43

5.6

42

5.4

41 41

0.85

(d) 50

6.8

5.2 40

0.7 0.75 0.8 Total mass (Kg)

42

43 44 45 46 47 Natural frequency (Hz)

48

49

50

40 0.089

0.090

0.091

0.092 0.093 Compactness

0.094

0.095 0.096

Figure 11. Relationship between different design objectives: (a) the relationship between mass and stiffness; (b) the relationship between mass and natural frequency; (c) the relationship between natural frequency and stiffness; (d) the relationship between compactness and natural frequency.

because each particle in one iteration evaluates only one time for the objective functions. For the RMOSA, however, the number of objective function evaluations is increased dramatically when the iteration times is increasing. The number of objective function evaluations for the RMOSA is always larger than that of PSO under the same iteration times. For the testing given in Figures 7 and 8, we can see that the regular PSO convergent after 602 iteration times while RMOSA convergent after only 20 iteration times, via the comparison shown in Figure 9, the RMOSA also requires fewer number of objective function evaluations, thus the proposed RMOSA has higher efficiency of convergence.

Simulation experiments In this section, the MOD problem is tested under the proposed algorithm. Suppose that all the materials are stainless steel, the density is  ¼ 7:9  103 kg=m3 , the

elastic modulus is E ¼ 2:06  106 MPa, the Poisson’s ratio is ¼ 0:3. The design  space is given as X ¼ a, Rc1 , rc1 , Rc2 , rc2 , Rp1 , rp1 , Rp2 , rp2 . Then the MOD problem can be formulated as follows min WðXÞ, ðXÞ, 1=KðXÞ, 1=!ðXÞ s:t: ji j4420 MPa, i ¼ 1, 2, . . . , 24 a 2 ½0:1, 0:15m, Rc1 , Rc2 , Rp1 , Rp2 2 ½0:003, 0:005m, rc1 , rc2 , rp1 , rp2 2 ½0:001, 0:004m,

ð17Þ

By applying the RMOSA, the Pareto optimal front of the design parameters can be obtained, as given in Table 1. The iteration history of the design objectives is as shown in Figure 10. By recording the trajectories of the objectives, one can obtain the relationship between different design objectives, as shown in Figure 11 and the relationship between design objectives and the

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Proc IMechE Part G: J Aerospace Engineering 227(11)

Stiffness (N/m)

8

5 x 10

(b) 45

7

40

6

35

Natural frequency (Hz)

(a)

5 4 3

25 20 15

2 1 0.1

0.11

0.12 0.13 0.14 Length of link a (m)

0.15

10 0.1

0.16

0.11

0.12

0.13

0.14

0.15

0.16

Length of link a (m) 5 (d) 8 x 10

(c) 45 40

7

35

6 Stiffness(N/m)

Natural frequency (Hz)

30

30 25

5 4

20

3

15

2

10 3

3.2

3.4

3.6

3.8

4 4.2 R1 (m)

4.4

4.6

4.8 5 x 10-3

1

3

3.2

3.4

3.6

3.8

4 4.2 R1(m)

4.4

4.6

4.8

5-3

x 10

Figure 12. Relationship between different design objectives and design parameters: (a) the relationship between a and stiffness; (b) the relationship between a and natural frequency; (c) the relationship between R1 and natural frequency; (d) the relationship between R1 and stiffness.

design parameters is shown in Figure 12. Then the designers can either select one optima solution from the Pareto optimal set as the practical design parameters, or find a compromise solution by bargaining different objectives based on Figures 11 and 12. From the Pareto optimal set, we can see that the length of the long link a are all 0.1 m, which is the minimum boundary value for the specified interval, this shows that shorting the design parameter a could lead to the better design objectives, the packaging/expansion ratio  is not sensitive to the changing of all design parameters in the given interval. Besides, the outer radius Rc1 for the center module are all near 0.005 m, which is the maximum boundary value for the given interval, this fact shows that the increasing the Rc1 leads to better natural frequency, higher stiffness and lower weight. To show that the reached results are Pareto optimal set, all objectives for the 21 particles at the finally value

and their corresponding previous iteration are shown in Figure 13.

Experiments Based on the optimized parameters given in the previous section, we select the first row of the Pareto optima set given in Table 1 as the parameters for the physical prototype, the optimal parameters are given in Table 2. The double-layer deployable mechanism has been designed and fabricated as shown in Figure 14. To evaluate the performances of the fabricated prototype, we have set up the stiffness and vibration measurement experiments. The stiffness in the given point of the mechanism measures the displacement under specified loads. We can measure the stiffness by first exerting force on a given direction in the testing point, then we measure its displacement under this given force. The experimental equipments are as

Huang et al.

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(b)

2

60

Pareto optimal Previous iteration

1.8

Pareto optimal Previous iteration 55

Natural frequency (Hz)

Total mass (Kg)

1.6 1.4 1.2 1

50

45

40

0.8 35

0.6 0.4

(c)

30

0

5

10 15 Sample points

20

25

5

11

0

5

10 15 Sample points

20

25

(d) 0.0975

x 10

Pareto optimal Previous iteration

Pareto optimal Previous iteration

10

0.097 9

Compactness

Stiffness (N/m)

8 7 6

0.0965

0.096

5 4

0.0955

3 2

0.095 0

5

10 15 Sample points

20

25

0

5

10 15 Sample points

20

25

Figure 13. The Pareto optimal set of design objectives for all the sample points and the previous iterations: (a) the Pareto optimal set of total mass; (b) the Pareto optimal set of natural frequency; (c) the Pareto optimal set of stiffness; (d) the Pareto optimal set of compactness.

to the Pareto optima set as given in Table 1, we can see that initially, the stiffness in point A is lower than the one obtained from the proposed algorithm, but p p p p c c c c a(m) R1 (m) r1 (m) R2 (m) r2 (m) R1 (m) r1 (m) R2 (m) r2 (m) the final point C is larger than the theoretical value. 0.1 0.005 0.0025 0.0027 0.002 0.005 0.004 0.005 0.0044 The stiffness in point B is closed to the theoretical value. From the experimental results, we can see that in the shown in Figure 15, the stiffness for a peripheral point is beginning stage, the stiffness is lower than the theoretby using the force sensor and the displacement sensor, ical value. This is because the existence of the joint the measured results are shown in Figure 15(b). For the tolerances in the mechanism. Also, the mechanism is initial testing stiffness, point A in Figure 15 (b), the value not stable rigid structure in this stage. Once the press of stiffness can be calculated as force is large enough, the displacement under the press force become small, and the stiffness of the mechanism Force 65 is close to the theoretical value. Therefore, preload to ¼ K1 ¼ ¼ 4:643  104 kN=m 3 stiffen the devices is also an important consideration Displacement 1:4  10 ð18Þ in the deployable mechanism design, so that the mechanism becomes stable structure in the deployed configuration. The vibration performance is also evaluated by meaThe stiffness in other point in Figure 15(b) can be calculated similarly, as shown in Table 3. By comparing suring the amplitude of vibration under different Table 2. Optimal parameters for the prototype.

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Proc IMechE Part G: J Aerospace Engineering 227(11)

Figure 14. The fabricated double-layer deployable mechanism: (a) deployed configuration; (b) mid configuration; (c) folded configuration.

(a)

(b) 1.4 1.2

Pressure transducer

C

1 0.8

B

0.6 0.4 0.2 0

Displacement sensor

0

A

50

100

150

200

250

300

350

400

Figure 15. The stiffness measuring for the double-layer deployable mechanism: (a) the experimental equipment; (b) the displacement along Z-axis under different loading force.

Table 3. The stiffness along Z-axis under different loading force. Position

Point A (kN/m)

Point B (kN/m)

Point C (kN/m)

Stiffness

4:643  104

4:3737  105

1:6  106

excitation frequencies, by using the frequency adjustable vibration exciter as the source of the vibration, the vibration performance can be evaluated by the amplitude under the vibration, as shown in Figure 16(a), two points are tested, one located at the performance base (point A in Figure 16(a)) and the other located at the connection point (point B in Figure 16(a)) of the two long links. The amplitude of

vibration under different excited frequencies is shown in Figure 16(b). The optimal design parameters show that in order to reach better stiffness and natural frequency, the radial parameters of the center modules are different from peripheral modules, therefore, efficient search algorithm in the design of this kind of mechanism is necessary. While the vibration frequency is increasing, the amplitude of vibration reaches its peak value at the frequency of 43 Hz, this is the first-order natural frequency of the mechanism. In the vibration test, amplitudes of vibration for two points are measured, as shown in Figure 16(b), the two points have the same peak-value frequency, but with different amplitude value.

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Figure 16. The amplitude measuring under different excitation frequency: (a) the experimental equipment; (b) the amplitude of vibration in Z direction under different excitation frequency.

Conclusions In this article, a novel double-layer deployable mechanism is proposed. The MOD problem of the proposed mechanism was formulated by considering the total mass, stowage ratio, stiffness and natural frequency. Aiming at solving this complicated MOD problem, a randomized multi-objective searching algorithm was proposed using a swarm of the controlled random search algorithm as its searchers. The proposed algorithm was tested under typical tested function, the results show that the proposed algorithm has higher convergence efficiency by comparing to the traditional PSO algorithm. The design problem has been tested under the proposed algorithm, and the Pareto optima solutions have been obtained, the relationship between different objectives is figured out, from these figures, designers can select the compromise optimal solutions intuitively based on specific requirements. The relationship between different design objectives can also be analyzed based on the Pareto optima solutions. Finally, the physical prototype is fabricated and the stiffness and natural frequency experiments are conducted. The experimental results demonstrate that the proposed mechanism offers an attractive combination of performance characteristics, which also have good consistency with the optima results that have been calculated by the proposed algorithm. Funding This work is financially supported by the National Natural Science Foundation of China (Project No. 50935002, and 51175105). The work is also supported by Shenzhen Fundamental Research Fund (Project No. JC201105160555A).

Acknowledgment The authors would like to thank the editors and the reviewers for their valuable comments.

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