Research Article

Advances in Mechanical Engineering 2017, Vol. 9(9) 1–10 Ó The Author(s) 2017 Multi-objective optimization design of DOI: 10.1177/1687814017720053 cycloid pin planetary reducer journals.sagepub.com/home/ade

Yaliang Wang, Qijing Qian, Guoda Chen, Shousong Jin and Yong Chen

Abstract A multi-objective optimal model of a K-H-V cycloid pin gear planetary reducer is presented in this article. The optimal model is established by taking the objective functions of the reducer volume, the force of the turning arm bearing, and the maximum bending stress of the pin. The optimization aims to decrease these objectives and obtains a set of Pareto optimal solutions. In order to improve the spread of the Pareto front, the density estimation metric (crowding distance) of non-dominated sorting genetic algorithm II is replaced by the k nearest neighbor distance. Then, the improved algo- rithm is used to solve this optimal model. The results indicate that the modified algorithm can obtain the better Pareto optimal solutions than the solution by the routine design.

Keywords Cycloid speed reducer, planetary , multi-objective optimization, evolutionary algorithm, density estimation

Date received: 11 January 2017; accepted: 15 June 2017

Academic Editor: Ismet Baran

Introduction between cycloid disk and housing rollers affected the contact force, friction torque, and transmission effi- Cycloid speed reducers have some excellent characteris- ciency. Hsieh6 proved the nonpinwheel design of tics, such as compact structure, wide scope of ratios, cycloid speed reducer could effectively reduce vibra- high transmission efficiency, low noise, and smooth- tion, stress value, and stress fluctuation. ness. Thus, they are widely used in all kinds of mechan- In the engineering design, many multi-objective opti- ical transmission. However, many design parameters mization problems (MOOPs) can be found. Some or all and complex constraints in the reducer design bring objectives often conflict with each other; in other more difficulty. At present, a lot of researches are avail- 1 words, all objectives cannot achieve its own best value able in this area. Chen et al. established the equation simultaneously.7 Thus, the policymakers need to choose of meshing for small teeth difference planetary gearing the compromise design parameters according to the and a universal equation of cycloid gear tooth profile based on cylindrical pin tooth and given motion. He et al.2 carried out optimum design and experiment on Key Laboratory of Special Purpose Equipment and Advanced Processing the double crank ring-plate-type pin-cycloid planetary Technology, Ministry of Education, Zhejiang University of Technology, 3 drive to reduce its noise and vibration. Sensinger pre- Hangzhou, China sented a unified design method to optimize profile, efficiency, and stress. Blagojevic et al.4 Corresponding author: Shousong Jin, Key Laboratory of Special Purpose Equipment and introduced a new two-stage cycloid speed reducer, Advanced Processing Technology, Ministry of Education, Zhejiang which was characterized by good load distribution and University of Technology, 18 Chaowang Road, Hangzhou 310014, China. 5 dynamic balance. Blagojevic et al. found the friction Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 Advances in Mechanical Engineering reality. For the last decade and more, many classical 2. Pin gear (also called pinwheel). It is uniformly multi-objective evolutionary algorithms (MOEAs) were distributed over the circumferential direction. studied, such as the non-dominated sorting genetic The pin gear consists of the pin and pin sleeve. algorithm II (NSGA-II),8 the improved strength Pareto 3. Cycloid gear. To keep the static balance of the evolutionary algorithm (SPEA2),9 and the multi- input shaft and increase the loading capacity of objective particle swarm optimization (MOPSO).10 As the reducer, two identical cycloid gear structures an efficient algorithm, NSGA-II has been widely used are usually adopted. The cycloid are to solve MOOPs.11–13 Some researchers used NSGA-II installed on the dual eccentric sleeves, and their to optimize gear reducers. Deb and Jain14 proved position differs 180°. In order to reduce the fric- MOEAs can solve multi-speed gearbox design optimi- tion between eccentric sleeve and cycloid gear, zation problem with more than one optimal goal and the turning arm bearing is installed between the different kinds of design parameters. Tripathi and two parts. Chauhan15 applied NSGA-II to optimize the volume of 4. Output mechanism. This reducer often uses the gearbox and the surface fatigue life factor simultane- output mechanism that called pin axle type. ously. Sanghvi et al.16 used NSGA-II to solve optimiza- There are some cylindrical pin holes on the tion design problem of a two-stage helical , in cycloid gear for inserting the cylindrical pins which the bearing force and volume were optimized. (the cylindrical pin sleeve is mounted on the Three optimization methods (MATLAB optimum cylindrical pin). Thus, the rotation motion of tools, genetic algorithm, and NSGA-II) were com- cycloid gears can be output by cylindrical pins. pared. It was shown that NSGA-II obtained better optimization objective values than other methods. Li et al.17 combined NSGA-II and fuzzy set theory to Theoretical tooth profile of cycloid gear optimize dynamic model of steering mechanism and got As shown in Figure 2, rolling circle 2 is the circum- the better result than the original design. However, far scribed circle of base circle 1, and the center of base cir- too little attention has been paid to the research about cle 1 (Og) is also the origin of rectangular coordinate multi-objective optimal design of cycloid speed redu- system. According to the forming principle of cycloid cers. Yu et al.18 and Yu and Xu19 selected volume as gear profile, supposing the base circle 1 is fixed, the the optimization objective. Xi et al.20 selected volume locus of point M in circle 2 is a epicycloid when circle 2 and efficiency as optimization objectives, and the multi- scrolls from tangent point A to tangent point B. The objective optimization model was solved by transform- rotation angle of the circle 2 (with the center O) around ing it into a single-objective problem, which ignored the the circle 1 (with the center Og) is denoted by u. The 21 relationship between objectives. Wang et al. focused rotation angle of circle 2 is denoted by ub and the abso- on reducing the volume and improving the efficiency of lute angle of circle 2 is denoted by uh. The coordinates reducers, and the optimal model was solved by the of a point on the theoretic profile of cycloidal gear can MATLAB genetic algorithm toolbox. be expressed as In this article, a multi-objective optimization model of reducer is established to minimize the volume of the x = R sin u esinu 0 z h ð1Þ reducer, force of the turning arm bearing, and maximal y0 = Rz cos u ecosuh bending stress of the pin. It is expected to provide a new way for multi-objective optimization design of cycloid reducer and similar structural optimization problems. Moreover, in order to improve the spread of the Pareto front, the density estimation metric of NSGA-II is replaced by the k nearest neighbor distance.

Optimization design model of the K-H-V cycloid driver Figure 1 shows the typical structure of the cycloidal pinwheel transmission, which mainly consists of the fol- lowing parts:

1. Planet carrier. It is composed of two parts, namely, the input shaft and the dual eccentric sleeves. Figure 1. Typical structure of cycloidal pinwheel transmission. Wang et al. 3

Figure 2. Theoretical tooth profile of cycloid gear. Figure 3. Meshing force of pin gear and cycloid gear.

Fi is proportional to component of Du in the direction where Rz is the radius of the pin gear distributed circle, of Fi. That is e is the eccentric distance (e = ObOg =(K1Rz)=Zb), K1 is the short amplitude coefficient (K1 =(rb=Rz), rb is the li pitch radius of the pinwheel), and Zb is the tooth num- Fi}Ducosai = Du ð4Þ RZ ber of the pin gears (Zb =(uh=u)). Therefore, equation (1) can be expressed as where li is the vertical distance between Ob and the 9 direction of Fi. When li is equal to rb, and Fi is equal to K1 = x0 = Rz sin u Z sin Zbu Fmax. That is b ð2Þ K1 ; y0 = Rz cos u Z cos Zbu rb b Fmax } Du ð5Þ Rz According to the radius of curvature formula, the radius of theoretical tooth profile of cycloid gear can In the triangle ObBP, sin ui =(li=rb). According to be expressed as equations (4) and (5), Fi can be expressed as

3 li 2 2 (1 + K 2K cosu ) R Fi = Fmax = Fmax sin ui ð6Þ 1 1 b Z r r0 = 2 ð3Þ b K1(1 + Zb) cos ub (1 + ZbK1 ) The maximum engaging force22 is shown as

Meshing force of pin gear and cycloid gear 4Tg Fmax = ð7Þ K1ZgRz As shown in Figure 3, the angular velocity (vg)of cycloid gear has the opposite direction on output tor- where Zg is the tooth number of the cycloid gear. In que when pin gears are fixed. In the switching mechan- general, Tg = 0:55Tv, and Tv is the output torque. ism, the rotation direction of cycloid gear and pinwheel Therefore, Fmax can be expressed as are same. On the left side of the Y axis, pinwheels and cycloid gear mesh, and the direction of force (Fi) of the 2:2Tv pin gear on the cycloid gear intersect with node P.On Fmax = ð8Þ K1ZgRz the right side of the Y axis, pin gears leave cycloid gear, and there is no force between them. Supposing Tg rep- In the triangle PObOi, according to sine theorem, resents the resistance torque of each cycloid gear, when sin ui can be expressed as a torque with the equal value and opposite direction of T is applied on the cycloid gear, the center of pin gear Rz g sin ui = sin ubi ð9Þ has a tiny circumferential displacement Du. The size of POi 4 Advances in Mechanical Engineering

Besides, according to cosine theorem, POi can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 POi = Rz + rb 2Rzrb cos ubi ð10Þ

Because K1 is equal to (rb=Rz), POi can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 POi = Rz 1 + K1 2K1 cos ubi ð11Þ

According to equations (6), (8), (9), and (11), Fi can be expressed as Figure 4. Typical structure of a pin gear with two pivots.

2:2T sin u = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv bi 12 Fi ð Þ Maximal bending stress of the pin. The break in pin is one K Z R 1 + K2 2K cos u 1 g z 1 1 bi of the main failure forms of this cycloid reducer. Hence, minimizing the maximal bending stress of the pin is the third sub-objection. For pin gear which has two ful- Design variables crums (in Figure 4), the bending stress of the pin is A vector which consists of five design variables is shown as follows 0 0 expressed as X =(Dz, dz, B, K1, dw), where Dz is the dia- 0 Mw max 44L1L2Tv meter of the pin gear distributed circle, dz is the dia- s ’ = 15 F 0 3 0 3 ð Þ meter of the pin, B is the width of the cycloid gear, and 0:1d LK1ZgDzd 0 z z dw is the diameter of the cylindrical pin. 0 0 where L1 = 0:5B + d +0:5D, L2 = 1:5B+d +d+0:5D, 0 and L = L1 + L2 = 2B + 2d + d + D. D is the thick- 0 Objective functions ness of the side wall of a pin gear housing. d is the interval between cycloid gear and the internal face of For the convenience, the input power, input speed, 0 transmission ratio, turning arm bearing, and the num- the side wall, and in general, dz D B. The third sub- ber of cylindrical pins are given. Under these premises, objective is presented as the sub-objectives are presented as follows. 44L L T min f (X)= 1 2 v 16 3 0 3 ð Þ LK1ZgDzdz Volume of reducer. The radial dimension is affected by the diameter of the pin gear distributed circle (Dz) and the diameter of the pin sleeve. The axle dimension is Constraint conditions affected by the width of the cycloid gear (B) and the gap between two cycloid gears. Therefore, the volume Short amplitude coefficient. If the short amplitude coeffi- can be expressed as20 cient is bigger, the minimal radius of theoretical tooth profile of cycloid gear will be reduced, which will p 0 2 decrease the outer radius of the pin sleeve, that is, the min f1(X)= (Dz + dz + 2D1) (2B + d) ð13Þ 4 contact strength of the cycloid gear and the pin gear increases. According to equation (16), the bending where D1 is the thickness of pin sleeve and d is the gap between the two cycloid gears. In general, d = b B (b stress of the pin will increase with the decrease in the is the width of the turning arm bearing). short amplitude coefficient. It suggests that the con- straint range of the short amplitude coefficient is [0.45, 0.8];20 thus, the constraint equation can be defined by Radial load on turning arm bearing. The service life of turn- ing arm bearing largely depends on its radial load. It g1(X)=0:45 K1 0 ð17Þ further affects the service life of the speed reducer. Hence, minimizing the radial load of the turning arm g2(X)=K1 0:8 0 ð18Þ bearing should be considered. It can be expressed as Cycloid tooth profile. To prevent cycloid tooth profile 2:6TgZb min f2(X)= ð14Þ from undercut and sharp angle, the ratio of an external K1DzZg diameter of the pin sleeve to a diameter of the pin gear Wang et al. 5

sffiffiffiffiffiffiffiffiffiffi distributed circle should be less than the minimum coef- FiEd ficient of the theoretical tooth profile curvature radius sH = 0:418 ð26Þ Brd (amin). Accordingly, this constraint can be defined by

where Fi is the meshing force in a certain position (d0 + 2D ) z 1 between the pin gear and the cycloid gear, rd is the g3(X)= amin 0 ð19Þ Dz equivalent curvature radius of the contact point, and E is the equivalent elastic modulus between the pin where a =(1 + K )2=(1 + K + Z K ). d min 1 1 g 1 gear and the cycloid gear. Since both materials are 5 GCr15 bearing steel, Ed is equal to 2:10 3 10 MPa. Maximum diameter of the cylindrical pin hole. In order to The constraint can be defined by guarantee the strength of the cycloid gear, there must sffiffiffiffiffiffiffiffiffiffi be a certain thickness (T) between two adjacent cylind- F E g (X)=0:418 i d s 0 ð27Þ rical pin holes, and in general, T = 0:03Dz. Thus, the 8 Br HP maximum diameter of the cylindrical pin hole meets the d following constraints where sHP is the allowable contact stress.

g4(X)=2T Dw + dsk + D1 0 ð20Þ Bending strength of the pin gear. According to equation p g5(X)=T Dw sin + dsk 0 ð21Þ (15), the constraint is as follows Zw

44L1L2Tv where Zw is the number of cylindrical pins. Dw is the g (X)= s 0 28 9 0 3 FP ð Þ diameter of the cylindrical pin hole distributed circle, LK1ZgDzdz which can be defined by where sFP is the allowable bending stress.

dfc + D1 Dw = ð22Þ 2 Contact strength between the cylindrical pin and the cylindrical 22 where d is the diameter of the root circle of a cycloid pin hole. According to Rao, this constraint is as fc follows gear. D1 is the diameter of the cycloid gear center hole, which is also the external diameter of the turning arm sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bearing. 10K1TvDz g10(X)=0:0949 sHP 0 2 DzK1 dsk is the diameter of the cylindrical pin hole, which ZwDwB(rwZb + 2 rw) can be defined by ð29Þ

dsk = dw + 2e ð23Þ where rw is the radius of the cylindrical pin sleeve, which can be defined by where dw is the external diameter of a cylindrical pin sleeve. 0 d r = w + D ð30Þ w 2 2 Pin-diameter coefficient. In order to guarantee the D strength of the pin gear housing and avoid the interface where 2 is the thickness of the cylindrical pin sleeve. 22 of pin gears, the value of pin-diameter coefficient (K2) should be in the range of 1.25–4. Thus, this constraint Bending strength of the cylindrical pin. According to Rao,22 condition can be expressed as this constraint can be expressed as

g6(X)=1:25 K2 0 ð24Þ 96Tv(1:5B + d) g11(X)= sFP 0 ð31Þ Z R d0 3 g7(X)=K2 4 0 ð25Þ w w w where K =(D =(d0 + 2D )) sin (p=Z ). 2 z z 1 b Life of the turning arm bearing 10 106 C 3 g (X)=L 0 ð32Þ Contact strength of the cycloid gear and the pin gear. To pre- 14 h 60n F vent the tooth surface from scuffing failure and fatigue pitting, the meshing between the pin teeth and the where Lh is the turning arm bearing life, and Lh is usually cycloidal gear teeth should meet the contact strength. equal to 5000 h; n is the rotating speed of the bearing; C Using Hertz theory, the contact stress (sH) between the is the dynamic load rating; F is the equivalent dynamical pin gear and the cycloid gear can be expressed as load; and F = 1:2R (R is the radial load on bearing). 6 Advances in Mechanical Engineering

Table 1. Pseudocode of k nearest neighbor distance. k nearest neighbor distance

% chromosome consists of the decision variables, value of the objective functions and rank, and the rank of every individual is based on non-domination. % V is the dimension of decision variable space. % M is the dimension of the objective space. % population consists of the decision variables, value of the objective functions, rank and k nearest neighbor distance. obj = chromosome (:,V + 1:M + V); for i = 1: M obj(:,i) = obj(:,i)/(max(obj(:,i))-min(obj(:,i))); %normalize sub-objectives end for n = jjchromosome ; %number of solutions (individuals) in chromosome for j = 1: n–1 for m = j + 1:n Ej, m = kkobj(j, : ) obj(m, : ) 2; %the Euclidean distance between two individuals in objective space. E(m,j) = E(j,m); end for E(j,j) = 0; %E stores the k nearest neighbor distance of every individual end for SE = sort(E,2); %sort the distance in ascending order population = zeros(n, M + V + n); population (:,1: M + V + 1) = chromosome; population (:,M + V + 2:M + V + n) = SE(:,2:n);

Improved NSGA-II (NSGAN) Non-dominated ranking and crowding distance are adopted in NSGA-II to control the evolutionary popu- lation size. The density estimation technique of the crowding distance calculates the average distance of two points on either side of this point along each of the objectives. Sometimes, this method cannot completely reflect the crowdedness between individuals. As shown in Figure 5, supposing that d represents the unit of a distance in objective space, the distance of solution A (12d) is equal to that of the solution B (12d) according to crowding distance. And, it can also be seen that B is more crowded than A. However, the density estimation metric of the k nearest neighbor distance9 can avoid Figure 5. Pareto front. this case. Through the k nearest neighborpffiffiffi distance, the first nearest neighborpffiffiffi distance of B (2 2d) is smaller than that of A (3 2d), which means B is more crowded represents the number of the design variables. The two algorithms run 30 times independently. According to than A. Hence, the density estimation metric of the k 22 nearest neighbor distance is introduced to improve the Table 2 and Rao, the range of design variables is spread of Pareto front, and the improved algorithm is shown in Table 3. marked as NSGAN. The k nearest neighbor distance The optimization procedure is shown in Figure 6. At computation procedure is shown in Table 1. first, an initial population is randomly created in the range of design variables, and the solutions utilize real- Optimization design example number encoding. Second, compute the rank and k nearest neighbor distance for the every individual in GCr15 material is selected for pin, pin sleeve, cycloid population. Then, the binary tournament selection, gear, cylindrical pin, and cylindrical pin sleeve. The recombination, and mutation operators are used to cre- main model parameters22 are shown in Table 2. ate an offspring population. In a binary tournament NSGA-II and NSGAN are adopted to optimize this selection, a lower rank and bigger k nearest neighbor model, respectively. The parameters in algorithms are distance is the selection criteria. NSGAN uses simu- listed as following: population size N is 100, evolu- lated binary crossover and polynomial mutation. After tional generation is 600, crossover probability pc is 0.9, that, the next population is selected from the offspring and mutation probability pm is 1/n, where ‘‘n’’ and previously population based on the rank and k Wang et al. 7

Table 2. Model parameters.

Parameters Values

Input power (kw) 4 Input speed (r/min) 1440 Transmission ratio 29 Number of the cylindrical pins, Zw 10 Contact stress, sHP (MPa) 850 Bending stress, sFP (MPa) 150 External diameter of turning arm bearing, D1 (mm) 86.5 Width of turning arm bearing, B (mm) 25 Rated dynamic load of the turning arm bearing, C (N) 64,900

Table 3. Range of design variables.

Design variables Interval Unit

Diameter of pin gear distributed circle 200

nearest neighbor distance. If the maximal generation is reached, output the result; if not, keep on evolving.

Spread test There are three sub-objectives, and the spread indica- tor23 is chosen to measure the spread of the Pareto front corresponding to obtained solutions. The smaller this value is, the more uniform the solutions distribute. The indicator is defined as

Pm P d(Ei, O)+ X 2O d(X, O) d i = 1 D = Pm ð33Þ d(Ei, O)+(jjO m)d i = 1 Figure 7. Pareto front obtained by NSGA-II. where O is a set of solutions, (E1, ..., Em) are m extreme solutions in the set of Pareto optimal solutions, m is the Compute the spread indicators of the Pareto front number of objectives and obtained by two algorithms. The Pareto fronts obtained by the two algorithms are shown in Figures 7 and 8. d(X, O) = min kkF(X) F(Y) ð34Þ Y2O, Y6¼X Spread indicator statistical result is shown in Table 4. From Figures 7 and 8, it can be shown that the 1 X d = d(X, O) ð35Þ NSGAN gets a more uniform Pareto front than that X 2O jjO by NSGA-II. Table 3 shows a statistical result of the Three extreme solutions (i.e. these solutions have a spread indicator. These data demonstrate that NSGAN maximal sub-objective value) are selected among the obtains better result than NSGA-II in terms of the obtained solutions based on the two algorithms. Their spread. In order to prove NSGAN has a significant objective values are shown as follows: E1 (3.11e–3, role in the spread, a t-test is performed for the average 4746.33, 34.44), E2 (1.49e–3, 10756.94, 94.89), and E3 of the spread indicator by software SPSS. The signifi- (1.43e–3, 8444.82, 150.00). Among them, E1 and E2 cance level is supposed as 0.05. The t statistic is 53.513, come from NSGAN, and E3 comes from NSGA-II. and the corresponding two-tailed probability is 0.000. 8 Advances in Mechanical Engineering

Figure 8. Pareto front obtained by NSGAN. Figure 9. Non-dominated Pareto front obtained by NSGAN.

Table 4. Average and standard deviation of the spread indicator.

Algorithm Average Standard deviation

NSGA-II 0.6302 4.82e–2 NSGAN 0.1431 1.28e–2

Because the two-tailed probability is smaller than the significance level, the null hypothesis is refused. It shows that two algorithms have significant differences in spread indicator.

Results analysis Figure 10. Three sub-objectives are smaller than that of the A total of 100 non-dominated solutions are picked out routine design. from 3000 solutions gained by NSGAN according to rank and k nearest neighbor distance. The minimal sub- objective solutions (volume, radial load of the turning Through the above analysis, NSGAN can obtain arm bearing, or maximal bending strength of the pin) solutions for which the three sub-objectives are super- are selected out from 100 non-dominated solutions. ior to that of the routine design. However, in the prac- These solutions are Solution 1 (1.37e–3, 9188.71, tical design, due to some sub-objectives conflict with 149.93), Solution 2 (2.92e–3, 4741.87, 40.96), and each other, the designer has to make a compromise. Solution3 (2.82e–3, 4853.17, 33.50). From three extreme For example, the designers can minimize the volume of solutions, three sub-objectives cannot reach minimum the reducer and maximal bending stress of the pin, but together. Figure 8 presents the Pareto front of 100 non- appropriately increase the radial load of the turning dominated solutions. In addition, the sub-objectives of arm bearing under the constraints, i.e., make a compro- the routine design22 are also shown in Figure 9. mise. Depending on this situation, some preference Next, the solutions for which the three sub-objectives solutions are selected from the 100 non-dominated are all smaller than that of the routine design are picked solutions. For the better comparison, Figure 11 shows out from Figure 9, which are shown in Figure 10. There the relationship between volume and radial load of the are eight total solutions whose three sub-objectives are turning arm bearing, and Figure 12 illustrates the rela- smaller than that of the routine design. Table 5 presents tionship between volume and maximal bending stress the design variables and sub-objectives of these eight of the pin. When the volume increases, the radial load solutions and the routine design. Besides, the design of the turning arm bearing will be reduced. Moreover, variables are rounded under the premise of meeting the according to equations (14) and (15), the f1 (volume) constraints. and f2 (radial load of the turning arm bearing) are Wang et al. 9

Table 5. Design variables and sub-objectives.

Design method Design variables Sub-objectives 3 Diameter of Diameter of Width of Short width Cylindrical f1 (m ) f2 (N) f3 (MPa) 0 0 pin gear pin, dz (mm) cycloid gear, coefficient, K1 pin, dw (mm) distributed circle, B (mm) Dz (mm)

Routine design 240 10 17 0.75 20 2.16e–3 6322 80.38 NSGAN 233.313 9.526 12.000 0.798 21.726 1.80e–3 6112 78.08 Rounded values 233 9.5 12.0 0.80 21.7 1.80e–3 6105 78.57 NSGAN 242.837 9.469 12.000 0.800 22.297 1.94e–3 5858 76.07 Rounded values 243 9.5 12.0 0.80 22.3 1.94e–3 5854 75.33 NSGAN 232.986 10.501 12.350 0.787 19.916 1.83e–3 6207 61.38 Rounded values 234 10.5 12.4 0.79 20.0 1.83e–3 6156 60.98 NSGAN 251.541 9.447 12.000 0.800 20.545 2.07e–3 5655 73.90 Rounded values 252 9.4 12.0 0.80 20.5 2.08e–3 5645 74.78 NSGAN 244.282 9.913 12.000 0.800 19.079 1.97e–3 5823 66.73 Rounded values 244 9.9 12.0 0.80 19.0 1.96e–3 5830 67.04 NSGAN 239.075 11.332 12.932 0.799 21.988 1.96e–3 5958 48.66 Rounded values 239 11.3 13.0 0.80 22.0 1.96e–3 5952 49.07 NSGAN 252.159 10.355 12.000 0.799 20.482 2.10e–3 5649 57.48 Rounded values 252 10.4 12.0 0.80 20.5 2.09e–3 5645 56.77 NSGAN 251.534 11.170 12.002 0.798 22.283 2.10e–3 5670 46.99 Rounded values 252 11.2 12.0 0.80 22.3 2.11e–3 5645 46.44

Figure 11. Volume of the reducer and radial load of the Figure 12. Volume of the reducer and maximal bending turning arm bearing. strength of the pin. conflicting. The increase of volume can reduce the max- according to Figure 12. These five preference solutions imum bending stress of the pin to a certain extent. sacrifice the second sub-objective, but the first sub- However, the maximum bending stress of the pin is not objective is obviously improved. entirely dependent on volume, and equation (16) shows that short amplitude coefficient can also have impact on maximal bending stress of the pin. Conclusion In Figures 11 and 12, there are five magenta prefer- ence solutions. As shown in Figure 11, the second sub- A multi-objective optimization model of a cycloid pin objective (radial load on bearing) is bigger than that of gear planetary reducer with the goals of minimizing the the routine design, but the first sub-objective (volume) volume, the radial load on turning arm bearing, and the is smaller than that of the routine design. Meanwhile, maximal bending stress of the pin is considered in this the third sub-objective (maximal bending strength of article. The density estimation technique of NSGA-II is the pin) is also smaller than that of the routine design improved using the k nearest neighbor distance. The 10 Advances in Mechanical Engineering improved algorithm (NSGAN) is used to solve the multiobjective optimization. In: Proceedings of the evolu- multi-objective optimization design model. Although tionary methods for design, optimization and control with some sub-objectives conflict with each other, the applications to industrial problems, Athens, 19–21 Sep- MOEA can optimize these sub-objectives simultane- tember 2001, pp.95–100. Barcelona: CIMNE. ously. NSGAN obtains more uniform Pareto fronts 10. Coello CAC, Pulido GT and Lechuga MS. Handling than NSGA-II, and the Pareto front can present the multiple objectives with particle swarm optimization. IEEE T Evolut Comput relationship between sub-objectives, which is more con- 2004; 8: 256–279. 11. Pasandideh SHR, Niaki STA and Asadi K. Bi-objective venient for the designers to select design solutions. optimization of a multi-product multi-period three- echelon supply chain problem under uncertain environ- Declaration of conflicting interests ments: NSGA-II and NRGA. Inform Sciences 2015; 292: 57–74. The author(s) declared no potential conflicts of interest with 12. Brownlee AEI and Wright JA. Constrained, mixed- respect to the research, authorship, and/or publication of this integer and multi-objective optimization of building article. designs by NSGA-II with fitness approximation. Appl Soft Comput 2015; 33: 114–126. Funding 13. Kamjoo A, Maheri A, Dizqah AM, et al. Multi-objective The author(s) disclosed receipt of the following financial sup- design under uncertainties of hybrid renewable energy port for the research, authorship, and/or publication of this system using NSGA-II and chance constrained program- article: This work was supported, in part, by a grant from ming. Int J Elec Power 2016; 74: 187–194. Natural Science Foundation of Zhejiang Province of China 14. Deb K and Jain S. Multi-speed gearbox design using (Nos LY16G010013, LY17E050023, and LQ16E050012), multi-objective evolutionary algorithms. J Mech Design National High-Tech R&D Program of China (No. 2003; 125: 609–619. 2015AA043002), and National Natural Science Foundation 15. Tripathi VK and Chauhan HM. Multi objective optimi- of China (No. 71371170). zation of planetary gear train. In: Proceedings of the simulated evolution and learning-international conference, Kanpur, India, 1–4 December 2010, pp.578–582. Rhein- References land-Pfalz: DBLP. 1. Chen BK, Fang TT, Li CY, et al. Gear geometry of 16. Sanghvi RC, Vashi AS, Patolia HP, et al. Multi-objective cycloid drives. Sci China Ser E 2008; 51: 598–610. optimization of two-stage helical gear train using 2. He WD, Li X, Li LX, et al. Optimum design and experi- NSGA-II. J Optim 2014; 2014: 1–8. ment for reducing vibration and noise of double crank 17. Li H, Niu W, Fu S, et al. Multiobjective optimization of ring-plate-type pin-cycloid planetary drive. Chin J Mech steering mechanism for rotary steering system using mod- Eng 2010; 46: 53–60. ified NSGA-II and fuzzy set theory. Math Probl Eng 3. Sensinger JW. Unified approach to cycloid drive profile, 2015; 2015: 1–13. stress, and efficiency optimization. J Mech Design 2010; 18. Yu Y, Yu B, Chen JX, et al. Optimal design of cycloid 132: 024503. cam planetary speed reducer. J Harbin Inst Technol 2002; 4. Blagojevic M, Marjanovic N, Djordjevic Z, et al. A new 34: 493–496 (in Chinese). design of a two-stage cycloidal speed reducer. J Mech 19. Yu M and Xu CY. Optimization design of cycloid cam Design 2011; 133: 085001. transmission based on MATLAB. Mod Manuf Eng 2010; 5. Blagojevic M, Kocic M, Marjanovic N, et al. Influence 8: 141–143 (in Chinese). of the friction on the cycloidal speed reducer efficiency. J 20. Xi QK, Lu WQ and Cao SB. Optimized design of cycloid Balk Tribol Assoc 2012; 18: 217–227. reducer based on genetic algorithm. J Mech Transm 2014; 6. Hsieh CF. Dynamics analysis of cycloidal speed reducers 38: 87–90 (in Chinese). with pinwheel and nonpinwheel designs. J Mech Design 21. Wang J, Luo SM and Su DY. Multi-objective optimal 2014; 136: 091008. design of cycloid speed reducer based on genetic algo- 7. Chen G, Sun Y, Lu L, et al. A new static accuracy design rithm. Mech Mach Theory 2016; 102: 135–148. method for ultra-precision machine tool based on global 22. Rao ZG. Designing of mechanism for planetary gearing. optimization and error sensitivity analysis. Int J Nanoma- Beijing, China: National Defense Industry Press, 1994 (in nuf 2016; 12: 167–180. Chinese). 8. Deb K, Pratap A, Agarwal S, et al. A fast and elitist mul- 23. Wang YN, Wu LH and Yuan XF. Multi-objective self- tiobjective genetic algorithm: NSGA-II. IEEE T Evolut adaptive differential evolution with elitist archive and Comput 2002; 6: 182–197. crowding entropy-based diversity measure. Soft Comput 9. Zitzler E, Laumanns M and Thiele L. SPEA2: improving 2010; 14: 193–209. the strength Pareto evolutionary algorithm for