*Corr. Author’s Address: Sakarya University of Applied Sciences, Sakarya Vocational School, Turkey, fkorkmaz@subu.edu.tr 392
Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402 Received for review: 2024-01-10
© 2024 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2024-05-02
DOI:10.5545/sv-jme.2024.921 Original Scientific Paper Accepted for publication: 2024-05-30
The Use of Heuristic Optimization Techniques on RV Cycloid 
Reducer Design: A Comparative Study
Korkmaz, F. – Dereli, S. – Karayel, D. – Kolip, A.
Furkan Korkmaz
1,*
 – Serkan Dereli
1
 – Durmuş Karayel
2
 – Ahmet Kolip
2
1
Sakarya University of Applied Sciences, Sakarya V ocational School, Turkey 
2
Sakarya University of Applied Science, Technology Faculty, Turkey
In this study, heuristic optimization algorithms such as particle swarm optimization (PSO) and quantum particle swarm optimization (QPSO) 
have been used to optimize the weight of the cycloid reducer based on the values of some parameters. The algorithms were applied separately 
to the design problem, and their performances were compared. Initially, a basic design was performed using conventional approaches. 
Constraint conditions have been defined based on geometric features and the essential component of the reducer, the cycloidal gear, and 
then the existing basic design has been optimized in terms of size and weight. Finally, finite element analysis was conducted to validate 
the obtained parameter values and control the stresses in the contact areas of the cycloid gear profile. It has been determined that PSO 
reduced the weight of the basic design by 33.3 %, whereas QPSO reduced it by 33.8 %. Moreover, the results of the finite element analysis 
showed that the design obtained according to both optimization methods was safe in terms of the resulting stresses. Consequently, it has 
been demonstrated that the selected design parameters and constraint equations have been sufficient and that the basic designs could be 
enhanced using intuitive optimization methods.
Keywords: cycloid reducer, finite element analysis, optimization, heuristic algorithm
Highlights
• 	 The research provides comprehensive constraint conditions necessary to reduce the weight of cycloidal reducers.
• 	 It shows that PSO and QPSO algorithms are suitable optimization methods for cycloidal reducers.
• 	 It shows that optimization methods are an accurate method for improving uncertain design parameters.
• 	 It recommends the control of cycloidal reducers by finite element analysis after optimization.
0  INTRODUCTION
Cycloid reducers are important as a power transmission 
mechanism in sectors requiring high precision. They 
are characterized by their high efficiency, low noise, 
and compact size. Cycloid reducers are generally used 
in a variety of applications, including machine tools, 
robotics, and wind turbines. Studies on cycloidal 
reducers have increased, especially with advances in 
robotic technology, focusing on analytical definitions 
for profile modification, geometry verification by 
numerical analysis, and optimization studies. Several 
researchers have proposed innovative methods for 
profile modification in cycloid reducers, such as 
optimizing the tooth shape to improve performance 
and reduce backlash. Some publications dealing with 
the profile equation suggested a new profile equation 
due to the complexity of the modification equation 
and the inability to improve the obtained result [1] and 
[2]. In the study of Li et al. [2], taking into account the 
pressure angle distribution characteristics, meshing 
backlash, tooth tip, and tooth root clearances, the 
modified profile is obtained by shifting the theoretical 
tooth profile in the direction of the normal contact 
point. Xu et al. [3] examined the effect of design 
parameters on load distribution and contact stresses. 
When creating profile equations, gaps and assembly 
errors must also be taken into account [4] to [7]. 
Yang et al. [4] developed the mathematical model of 
the closed-loop mechanism of the cycloid reducer. 
Manufacturing and assembly errors were added to the 
analytical model, and their effects were examined. 
Korkmaz and Karayel [8] studied on a prestressed 
system to prevent clearance between the cycloid gear 
and the pin. Another important criterion for profile 
design is the characteristic of the load distribution. It 
is expected that the load will be distributed on half of 
the pins and will change curvilinearly [9] to [11]. Li et 
al. [10] investigated the effects of radial and angular 
defects in ring pins on load distribution and maximum 
contact stress.
Numerous researchers used numerical analysis 
techniques to evaluate and optimize key parameters 
such as contact stress, efficiency, backlash, and 
dynamic behaviour to achieve optimal performance 
in cycloid reducers. Matejic et al. [12] observed the 
internal stresses on the cycloid gears, which is the 
most important element of cycloid reducers. For 
this purpose, they made a finite element analysis 
of the model they created and compared it with the 
experimental results. It was emphasized that the 
experimental and simulation results were quite close 
as an important condition in the results obtained. 
Also, this situation shows that it is appropriate to use 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
393 The Use of Heuristic Optimization Techniques on RV Cycloid Reducer Design: A Comparative Study
simulations in cycloid reducers. Jiang et al. used the 
finite element analysis method to verify the effect 
of the profile equation they created on the load 
distribution characteristic [1].
Optimization techniques were widely used in 
the field of cycloid reducer design to enhance their 
performance and efficiency [13] to [15]. Zhang et al. 
[16] conducted an optimization study on the carrying 
capacity of crank bearings in rotate vector (RV) 
reducers. In the study, a multi-objective genetic 
algorithm was used, and the results were confirmed 
by finite element analysis. Wang et al. [17] also 
proposed a methodology to maximize efficiency 
while minimizing the volume of the reducer using a 
genetic algorithm. Zhang et al. [18] determined three 
objective functions: high tensile, high efficiency, 
and low lightness parameters and thirteen boundary 
conditions. They used the non-dominated searching 
genetic algorithm (NSGA-II) as the optimization 
method. In the results, it was stated that as a result 
of the optimization, efficiency was increased, the 
weight was reduced, and the design parameters were 
provided. Huang et al. [19] focused on a more specific 
area. In their work, they carried out an optimization 
study to improve the profile of the bearings and 
crowned roller profile. The Crow Search Algorithm 
(CSA) was used, and it was stated that the bearing life 
increased in the results obtained.
Optimization is crucial in the design of cycloidal 
reducers, as in other types of reducers. Cycloidal 
reducers are generally optimized for efficiency, 
durability, size and weight, cost, vibration, and noise 
control. Size and weight significantly impact other 
variables that need optimization in the design of these 
reducers. Indeed, the optimization of size and weight 
plays a vital role in meeting performance, efficiency, 
economic, and overall system requirements in reducer 
design. Therefore, this study focuses on weight 
optimization.
Changes in the design parameters of the cycloidal 
gear, the fundamental component of cycloidal 
reducers, directly affect the dimensions of other parts 
of the reducer. Simultaneously, alterations in the 
design parameters lead to a change in the profile. The 
strength and performance characteristics of cycloidal 
reducers are determined by the cycloidal gear profile. 
Considering existing literature studies, it has been 
decided to optimize the determined design parameters 
by accurately defining constraint equations, aiming to 
minimize weight.
Reviewing optimization studies on cycloidal 
reducers, it is observed that, typically, reliance is 
placed on a single algorithm. However, it is evident 
from the literature that different optimization 
algorithms yield different results for the same 
objective. Certain algorithms demonstrate superior 
performance under specific conditions, while others 
might not deliver the same level of effectiveness. This 
is attributed to the fact that optimization algorithms 
might perform better for certain types of problems or 
conditions. The initial effectiveness of an algorithm 
on a particular problem is not always apparent. 
Despite the use of multiple algorithms in the literature, 
particle swarm optimization (PSO) stands out due to 
its ease of application, broad applicability, parallel 
processing capability, less parameter sensitivity, rapid 
convergence, and flexible parameter adjustment. 
Hence, in this study, the PSO and a version based 
on quantum mechanics principles, namely quantum 
particle swarm optimization (QPSO), have been 
utilized. Subsequently, three different designs 
obtained through the PSO, QPSO algorithms, and a 
general design approach have been compared against 
each other.
The designs obtained from both optimization 
algorithms have clearly shown better results in 
terms of weight compared to the basic design, as 
explicitly observed in the Result and Discussion 
section. Comparing the performance of two different 
algorithms and demonstrating the suitability of 
the optimization for the problem stands out as an 
originality of this study. For each optimization 
result, a new design has been executed, analysed, 
and compared with the basic design. Consequently, 
besides validating the optimization algorithm, 
factors not explicitly defined as constraints in the 
optimization, such as stresses, have been reviewed. 
This can be put forth as another differentiation of this 
study from other research in the field.
1 CYCLOID REDUCER DESIGN AND PARAMETERS
The profile of the cycloid gear, which is the most 
important part of cycloid reducers, varies according to 
the eccentricity (e), the diameter of the pins' locating 
circle (D
z
), and the diameter of the pins (d
z
). Design 
variables in cycloid gears also affect other equipment 
in the gearbox geometrically. Because of this situation, 
it directly affects the weight of the reducer and the 
efficiency of the system. 
1.1  Geometry of Cycloid Reducer
RV cycloidal reducers have two stages: planetary 
and cycloidal. The first stage involves a planetary 
mechanism (Fig. 1 (1)), which offers several 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
394 Korkmaz, F. – Dereli, S. – Karayel, D. – Kolip, A.
significant advantages. Firstly, the planetary 
mechanism directly influences the transmission ratio, 
allowing for precise control of the output speed. 
Secondly, it efficiently distributes the torque from 
the input shaft to the cranks (Fig. 1 (5)), which then 
transfers the torque to the second stage, ensuring 
smooth and reliable power transmission. It consists of 
the cycloid gear(s) (Fig. 1 (7)), which is the key part 
of the second stage reducer of cycloid reducers. In 
this study, the geometry of the cycloid gear has been 
optimized, especially considering the contact stresses 
and weight. The exploded view of the cycloid reducer 
and assembly used in the study is given in Fig. 1.
2  OPTIMIZATION TECHNIQUES
Nowadays, optimization methods are frequently 
used by researchers to solve complex and non-linear 
problems. Because classical numerical methods 
approach problems in a certain way, they produce very 
unsuccessful results in obtaining the optimum value of 
the problem [20]. For this reason, heuristic techniques 
inspired by the behaviour of creatures in nature have 
become very popular among researchers. The fact that 
they can produce a large number of optimum values, 
especially in many systems with multiple parameters, 
has further increased the fame of these techniques [21]. 
These techniques have been able to produce effective 
solutions to problems in many different fields within 
mechanical engineering with a single fitness function 
[22]. In one study, a new method was proposed based 
on the wolf algorithm, which is considered an up-
to-date heuristic algorithm, in order to estimate the 
rotation error of the RV reducer in the most accurate 
and fastest way [23]. In their work, they used seven 
different up-to-date heuristic algorithms to achieve the 
Fig. 1.  RV Cycloid reducer structure; (1) planet, (2) main bearing, (3) angular bearing, (4) front cover, (5) crank, (6) crank bearing,  
(7) cycloid gear, (8) case, (9) pins, and (10) rear cover
design of a two-stage gearbox with optimum values in 
terms of volume [24]. In this study, a cycloid reducer 
design with the most optimum values in terms of 
weight and size has been performed. The optimum 
values of the variables of the gear have been obtained 
separately with both standard PSO and quantum PSO 
techniques and compared in terms of results.
Classical PSO is an algorithm that has been 
widely studied in the literature, is accepted as the 
starting algorithm of heuristic techniques, and can be 
easily applied to the problem due to the small number 
of control parameters. Therefore, it is an important 
algorithm whose usability and ability to produce 
solutions to problems have been proven many times 
[25]. The PSO algorithm, which shows the power of 
the swarm effect in reaching a solution by a swarm of 
birds or fish moving together while reaching a target, 
is depicted in Fig. 2 in terms of the movement of the 
particles [26]. As can be clearly seen from this figure, 
all of the particles are moving towards the target at a 
certain speed:
1. Randomly initial values in the appropriate range
2. Loop iteration
3.    Loop particles
4.  Calculate the fitness function  
 (According to Section 2.2)
5.  Find the value of Pbest
6.  Update particle velocity value
7.  Update particle position
8.  Detection of constraint situations  
 (According to Section 2.3)
9.    End particle loop
10.    Find the value of Gbest
11. End iteration loop
12. Return Gbest value

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
395 The Use of Heuristic Optimization Techniques on RV Cycloid Reducer Design: A Comparative Study
Fig. 2.  Movements of particles in standard PSO and quantum PSO 
techniques
As seen in the pseudo-code expressing the flow 
of the particle swarm algorithm above, the only 
parameter in the algorithm is the velocity value 
calculated in line 6. Since this value is the particle's 
approach parameter to the target, it is recalculated for 
each particle in each iteration. Therefore, the particles 
approach the target at a certain distance based on 
Pbest and Gbest values. In this flow, the only step that 
changes from problem to problem is the line where 
the fitness function calculation is made in step four. 
Other steps remain largely the same.
The Quantum PSO technique, which is similar 
in name to the classical PSO algorithm, has come to 
the fore because the particles have a more dynamic 
structure, bypassing local convergence situations and 
guaranteeing the global optimum value [27]. In fact, 
Fig. 2 clearly depicts the movement of particles in both 
PSO and Quantum PSO algorithms. According to this 
figure, it is clear that all particles in the PSO algorithm 
are moving in the same direction. This situation is the 
biggest obstacle to a particle that is already lagging 
behind in reaching the optimum value and causes the 
algorithm to frequently get stuck at the local optimum 
value [28]. In Quantum PSO, inspired by quantum 
wavelet motion, all particles have the ability to move 
in any direction, according to this figure. Therefore, a 
particle lagging behind can easily reach the optimum 
value. This prevents the algorithm from being stuck in 
local best values and ensures that it obtains the global 
best value [29].
1. Randomly initial values in the appropriate range
2. Loop iteration
3.    Compute MBest (Average of Pbest Particles)
4.    Loop particles
5.  Calculate the fitness function
6.  Calculate quantum wavelength
7.  Update particle position
8.  Detection of constraint situations 
  (According to Section 2.3)
9.    End particle loop
10. End iteration loop
11. Return Gbest value
As seen in the pseudo-code of the Quantum PSO 
algorithm above, wavelength theory and the average 
of local best values are used for the next position of 
the particle. Of course, as in all stochastic algorithms, 
randomness is very important in heuristic algorithms 
that search in a certain solution space. In both the 
standard PSO algorithm and the Quantum PSO 
algorithm, this is effective in determining the new 
position of the particles. However, it is clearly seen in 
the literature that this alone is not sufficient in terms 
of the PSO algorithm [30]. 
In this study, the minimization of the weight and 
dimensions of a cycloidal reducer has been considered 
as an optimization problem. The objective functions, 
design variables, and constraints defined for the 
optimization problem are provided below.
2.1  Design Variables
One of the most important parameters for reducers 
is the transmission ratio. This ratio is calculated 
according to the relative transmission ratio formula for 
RV reducers. The relative transmission ratio formula 
for the RV reducer is as follows. [31] and [32]:
. i
z
z
z


1
1
3
4
2
, (1)
here, z
3
 and z
4
 are the tooth numbers of the sun gear 
and planet gear, respectively. They are taken equally 
because of the focus on the cycloid disk geometry. 
The reducer transmission ratio was preferred to be 
1/40. For this reason, the number of lobes in the disks 
(z
1
) is 39, and the number of pins (z
2
) is 40. 
A new geometry has been designed considering 
the design parameters in Table 1 for the sample 
application. The parameters were decided according 
to the 1/40 transmission ratio.
The eccentricity (e), the diameter of the pins' 
locating circle (D
z
), the diameter of the pins (d
z
) 
and the thickness of the cycloid gear (B) have been 
accepted as the design parameters. These parameters 
also affect the geometry of other parts that make up 
the gearbox:
 X = [e, D
z
, d
z
, B]. (2)
The lower and upper limits of the optimization 
parameters given in Table 1 were chosen close to the 
values in articles and catalogues containing previous 
successful application examples. Thus, it is aimed 
to protect against possible risks that may arise in 
the material supply, manufacturing, assembly and 
operating performance of the reducer parts.

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
396 Korkmaz, F. – Dereli, S. – Karayel, D. – Kolip, A.
Table 1.  Design variables and values for cycloid reducer
Parameter
Symbol, 
unit
Initial 
value
Upper and  
lower limits
Eccentricity e [mm] 1.5 0.8 ≤ e ≤ 2
Diameter of the pins'  
locating circle
D
z
 [mm] 154 110 ≤ D
z
 ≤ 160
Pin diameter d
z
 [mm] 6 4 ≤ d
z
 ≤ 8
Thickness of cycloid gear B [mm] 11 6.4 ≤ B ≤ 12.8
2.2  Objective Function
When calculating the weights of the cycloid gears, 
which form the main part of the reducer, the cycloid 
gear e, D
z
, d
z
, B profile and the spaces inside are taken 
into account. The main dimensions of the cycloid gear 
are determined by other parts that make up the crank 
and front-rear cover fastening spaces. For this reason, 
when calculating the cycloid gear weight, the weights 
of other parts are usually included in the objective 
function of the optimisation study. These parts are 
cycloid gear (G
1
), case (G
2
), crank (G
3
) and pin (G
4
). 
Also, we defined some coefficients in the objective 
function. We have 2 cycloid gears, 4 same-size planet 
gears, 3 cranks and 40 pins:
 G = 2×G
1
 + G
2
 + 3×G
3
 + 40×G
4
. (3)
2.2.1  Weight of Cycloid Gear
When calculating the weight of the cycloid gear, 
internal spaces are subtracted from the mean disc 
diameter (D
b
). These spaces are the spaces opened for 
the input shaft (D
s
), cranks (D
u
) and front-rear cover 
connections (D
k
). These terms are given in Fig. 3.
GB rdrd Dz DD
r
xy
sk uk
cc
1
0
2
0
22 2
22
4
3 














() () , , (4)
where K
ez
D
z
1
2
2


/
, and 
SK Kz   12
1
2
11
-c os()  is the meshing phase 
angle. The x
c
 and y
c
 are the coordinates for meshing 
the points where the pin and the disk come into 
contact. The x
c
 and y
c
 coordinates are expressed as in 
Eq. (5):
 
x
dK z
S
y
Dd
Kz
S
c
z
c
zz

 








2
22
1
11
11
sin( )
(- cos)
,


 (5)
Fig. 3.  Cycloid gear parameters
Fig. 4.  Case parameters
  
G
DH DH Dd HD H
zz zz z
2
13
2
3
2
1
22
23
2
3
4
22 24 02

       

  ()   




    




DH
DD H
z
zz
22
2222 2
35
2
4
123
2
13
2

 
5 5












c
.
  (6)

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
397 The Use of Heuristic Optimization Techniques on RV Cycloid Reducer Design: A Comparative Study
2.2.2  Weight of Case
The case varies according to the cycloid gear 
dimensions. Parameters of the case are given in Fig. 4.
2.2.3  Weight of Crank
The dimensions of the cranks vary according to the 
cycloid gear geometry and the torque value expected 
from the system.
Fig. 5.  Cranks parameters
 
Gd BD LL
qLcr 3
22
12
4
2    




 () .
 (7)
2.2.4  Weight of Pins
The diameters of the pins (d
z
) directly affect the 
cycloid gear profile. In addition, the lengths of the 
pins (l) are usually 0.5 mm more than the cycloid 
gear’s thickness.
 Gd l
zp in 4
2
4


 . (8)
2.3  Constraint Conditions
In optimization applications, accurate constraint 
conditions must be identified, and boundaries should 
be framed to ensure reliable results. To achieve this, 
constraint conditions commonly used in the literature 
have been evaluated and adapted for our specific 
problem.
2.3.1  Bearing Clearances in the Cycloid Gear
The outer raceway (D
m
 + D
we
) is situated between 
the root circle (D
c
) and the central hole (D
s
) of the 
cycloidal gear. Consequently, the bearing pitch 
diameter must be less than the gap between the 
diameter of the central hole and the diameter of 
the root circle. However, the diameters (D
L
) of the 
sections of the cranks mounted with the planet are 
related to the crank diameters and eccentricity [19]: 
 
gx DD
DD
gx DD De
mw e
cs
Lm we
1
2
2
0
20
   


    





. (9)
2.3.2  Distance Between Clearances
Another important constraint for RV reducers is the 
thickness of the outer ring of the eccentric bearings. 
Although the eccentric bearings used in RV reducers 
do not use an outer ring, the gap between the crank 
clearance and the tooth root diameter, as well as the 
gap between the crank clearance and the centre hole. 
Therefore, the distance between the clearances on 
the cycloid gear must not be less than the required 
thickness of the outer ring. The required thickness 
of the outer ring is obtained by multiplying the roller 
diameter by the ring thickness coefficient ( ε) [19] and 
[33]:
 
gx
DD D
DR
gx
DD
RD
D
mw es
we
mw e
we
c
30
40
22
0
22
0
 

 
 

 






 


. (10)
Here, R
0
 is the radius of the distribution circle of 
the cranks relative to the centre of the cycloidal gear. 
The ring thickness coefficient varies between 0.3 and 
0.5. In our problem, we have taken the value of 0.4 
[19] and [34].
2.3.3  Bearing Rollers
In addition to the crank and cycloid gear dimensions, 
the dimensions of the bearing rollers located between 
the cycloid gear and the crank are also very important 
for the life of the reducer. Therefore, a general 
approach regarding the bearing rollers has been taken 
into consideration [19]:
 gx LB
we 5
0    , (11)
where L
we
 is the length of the bearing roller.
2.3.4  Width of Cycloid Gear
The width of the cycloid gear directly influences not 
only the structure of the body and the length of the 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
398 Korkmaz, F. – Dereli, S. – Karayel, D. – Kolip, A.
pins but also surface stresses. Therefore, it is of great 
importance [17].
2.3.5  Tooth Profile Structure
In addition to the damage that may arise from the 
surface hardness of the tooth profile, the fractures 
that may occur at the tooth root due to the profile 
structure should not be overlooked. When forming 
the tooth profile, the structure of the profile should be 
constrained with the given constraint below [14]:
 gx
d
D
K
KZ K
z
z
8
1
2
11 1
1
1
0  



()
. (13)
2.3.6  Modification Coefficient for Cycloid Gear
The modification coefficient leads to different profiles 
in cycloid gear. Therefore, it directly affects the 
contact stresses occurring in the cycloid gear [17]:
 
gx K
gxK
91
10 1
06 50
09 0
  
 





.
.
. (14)
2.3.7  Diameter of the Pins' Locating Circle
The diameter of the reference circle where the pins 
are placed directly affects the dimensions of the body, 
crank, and cycloidal gear, as well as their profiles [16]:
 
gx TD
gxDT
in z
zi n
11
3
12
3
17 0
26 0
  
  





.
.
. (15)
2.3.8  Contact Strength Between Cycloid Gear and Pin
The most critical areas in cycloidal reducers are where 
pin-cycloid gear contact occurs. Therefore, during 
optimization processes, contact stresses between 
the pins and the cycloid gear must be specified as 
constraint conditions. Eq. (16) is defined for the 
constraint condition of the pin-cycloid gear contact 
stress [17]:
gx
E
B
T
KzDz
e
e
out
HP 13
1
0 418
44
1
0  



 .
.
.
min

 (16)
The curvature radius ( ρ
e
min
) is calculated by the 
following two equations [35]:
 


e
iz
iz
d
d
min
/
/
. 


2
2
 (17)
The material selected for both the pin and the 
cycloid gear is 20MnCr5 (1.7147). Heat treatments 
are applied to harden the surface of cycloid gears. 
Because of that, the maximum allowable contact 
stress ( σ
HP
) value is more than the other steels. The 
mechanical properties are given in Table 2 [36].
Table 2.  Mechanical properties of 20MnCr5
Maximum allowable 
contact stress, σ
HP
 
[MPa]
Equivalent elastic 
modulus, E
e
 [MPa]
Yield strength  
[MPa]
1200 210000 700
3  RESULTS AND DISCUSSIONS
The design parameters obtained through PSO and 
QPSO algorithms, along with the rounded values of 
the optimization results, are provided in Table 3.
Table 3.  Initial value and PSO-QPSO algorithms rounding results
Parameter
Initial 
value
PSO 
rounding 
results
QPSO 
rounding 
results
Upper and  
lower  
limits
Eccentricity 1.5 1.1 1.25 0.8 ≤ e ≤ 2
Diameter of the pins' 
locating circle
154 110.6 135.3 110 ≤ D
z
 ≤ 160
Pin diameter 6 4.2 6.3 4 ≤ d
z
 ≤ 8
Thickness of cycloid 
gear
11 11 11.7 6.4 ≤ B ≤ 12.8
The total weights obtained from the optimization 
studies conducted using PSO and QPSO algorithms 
are presented in Fig. 6. As observed in Fig. 6, the goal 
of achieving approximately a fifty percent reduction 
in weight compared to the basic design has been 
accomplished using QPSO. 
Fig. 6.  Total weight comparison of designs
As seen in Fig. 7, the loads are distributed 
across half of the pins, taking the pin in the full grip 
position between the pins and the cycloid gear as 
the starting point. In the case of full engagement, 
the mechanism is theoretically assumed to have no 

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
399 The Use of Heuristic Optimization Techniques on RV Cycloid Reducer Design: A Comparative Study
 F
l
r
F
i
i
w
=
1
max
. (18)
Generally, in RV cycloidal reducers that operate 
with two cycloid gears, the torque value cannot 
be evenly distributed due to manufacturing errors. 
Therefore, in the approach found in the literature, 
calculations are performed considering that 55 % of 
the output torque (T
out
) is applied to one cycloid gear 
as if it were acting on the precision cycloid gear. 
In this study, the output torque value was taken as 
300000 Nmm. Consequently, the maximum contact 
force is calculated according to Eq. (19) [37]:
 F
T
ez z
out
max
.
. 


40 55
12
 (19)
Eq. (20) is used when calculating the contact 
forces at the contact points separately for each pin  
(i = 1, 2, …, 40):
 F
FD
D
ze Dze
i
z
z
z


  
max
 (/ ) sin
() - cos
.
2
4
2
2
2
2


 (20)
Eq. (20) has been used to calculate the load 
distributions for three designs based on the values 
in Table 3. The resulting load distributions are given 
comparatively in Fig. 8 as basic design, PSO and 
QPSO. When examining the load distribution under 
a torque of 300000 Nmm, it is observed that the 
design with the highest maximum load occurs in the 
QPSO, which also has the lowest weight among the 
designs. Additionally, another observation in the load 
distribution is that the distribution between the pins is 
more regular in the basic design compared to PSO and 
backlash and calculations are made according to the 
full engagement of the pins.
Fig. 7.  Load distribution on cycloid gear-pins
As seen in Fig. 7, the virtual centroid circle with 
a radius of r
w1
 for the cycloid gear and the virtual 
centroid circle with a radius of r
w2
 for the pins are 
always tangent at a single point. φ
i
 indicates the 
angular position of each pin relative to the reference 
y-axis. The normal contact points of all pin-cycloid 
gears converge at this tangent point. Contact forces 
at the pin-cycloid gear contact points are calculated 
according to Eq. (18). The maximum meshing force 
(F
max
) occurs at the pin-cycloid gear contact point 
where the force arm (l
i
) is maximum (l
max
 = r
w1
) [37]:
Fig. 8.  Comparative load distribution graphs

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
400 Korkmaz, F. – Dereli, S. – Karayel, D. – Kolip, A.
QPSO. This reduction in size creates an opposite trend 
with the load distribution.
The pin-cycloid disk contact stress constraint is 
defined in Eq. (14). However, a constraint equation 
cannot be defined for the stresses except for the 
contact surfaces of cycloid gears. The stresses that 
may occur in the gear body can only be controlled 
by finite element analysis. For this purpose, three 
different geometries were modelled in the ABAQUS 
program with the design parameters obtained. Fig. 9 
shows the stresses resulting from the finite element 
analysis of the Basic design, PSO and QPSO in the 
case of full engagement. The results obtained show 
that in all three designs, the maximum stresses in the 
cycloid gear occur at the contact points with the pin. 
In order to check that the optimization algorithms 
are suitable for cycloidal reducers, the cycloidal 
disc-pin contact surface should be checked. For this 
purpose, a path is drawn along the gear profile. An 
example of a path created to examine the effect of 
profiles on the stresses occurring on the cycloid gear 
surfaces is given in Fig. 10.
Fig. 10.  Cycloid gear path along the surface
In Fig. 11, Von Mises stresses along the path 
provided in Fig. 9 under 300000 Nmm torque are 
presented. It has been determined that the maximum 
stresses are 275.76 MPa in the basic design, 371.10 
MPa in PSO, and 273.04 MPa in QPSO. Considering 
the stresses, it is evident that the PSO geometry has 
the highest value. Considering the stresses on the 
surface and geometric features, QPSO geometry is 
safer than PSO.
Also, in Fig. 11 shows that the pressure does not 
change logarithmically along the cycloid gear, unlike 
the analytical solutions. This is because backlash 
cannot be identified in analytical solutions.
4  CONCLUSIONS
Cycloid reducers are widely used in various industrial 
sectors due to their ability to withstand high torque, 
low backlash, high positioning accuracy, resistance 
to wear and sudden impacts, and efficient operational 
performance. However, achieving their optimal 
design through conventional design approaches is 
quite challenging due to non-standard components in 
these reducers, particularly the cycloidal gear profile, 
and the direct interdependence of design parameters 
on each other.
In this study, two widely applied intuitive 
optimization algorithms, PSO and QPSO, are utilized. 
The primary aim is to minimize the weights and total 
costs of the reducer components without exceeding 
the safety stress limits. Proposed constraint equations 
are developed for weight and size optimization 
of a fundamental cycloidal reducer design. The 
results obtained from both optimization algorithms 
are compared with each other. The comparison of 
algorithm performance considers weight reduction 
and analytical load distribution. Subsequently, 
different reducers are designed using the design 
parameters obtained from each algorithm separately. 
Each resulting reducer design undergoes finite 
element analysis and is compared with others.
As a result, it is determined that the proposed 
constraint conditions and boundary values are 
appropriate for the weight reduction problem. 
Furthermore, it is established that the PSO algorithm 
reduces weight by 33.3 %, while the QPSO algorithm 
reduces it by 33.8 % compared to the basic design. 
a)    b)    c) 
Fig. 9.  Stress analysis of three different designs: a) basic design, b) PSO, and c) QPSO

Strojniški vestnik - Journal of Mechanical Engineering 70(2024)7-8, 392-402
401 The Use of Heuristic Optimization Techniques on RV Cycloid Reducer Design: A Comparative Study
Analyses reveal that maximum stresses occur at the 
contact surfaces between the cycloid gear and the pins. 
This finding validates the constraint conditions and 
boundary values used for defining clearances inside 
the cycloid gear. However, there is not yet a constraint 
condition defined for stresses occurring on the inner 
surface of the cycloid gear. For this reason, finite 
element analysis is currently an important verification 
tool for the products obtained from optimization 
algorithms.
In the future, further studies as a continuation of 
this research could explore the use of more intuitive 
algorithms and hybrid algorithms for enhanced 
optimization in this field. In addition, it is observed 
that the stresses in the cycloid disc body increase as the 
size decreases. For this reason, lifespan calculations 
should also be carried out in future studies.
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