*Corr. Author’s Address: University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lučića 5, 10002 Zagreb, Croatia, daniel.miler@fsb.hr 725
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734 Received for review: 2022-08-29
© 2022 The Authors. CC BY-NC 4.0 Int. Licensee: SV-JME Received revised form: 2022-10-26
DOI:10.5545/sv-jme.2022.349 Original Scientific Paper Accepted for publication: 2022-11-03
Multi-Objective Optimization  
of the Chebyshev Lambda Mechanism
Miler, D. – Birt, D.– Hoić, M.
Daniel Miler
*
 – Dominik Birt – Matija Hoić
University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Croatia
Walking mechanisms are a solution for cases in which wheels are not applicable, such as uneven or stepped surfaces and surfaces with 
obstacles. Furthermore, it is possible to tailor mechanism footpaths to expected working conditions through optimization. Thus, in this 
paper, a mechanism optimization process was proposed, focusing on single-leg performance. Numerical Simulink calculations were used to 
determine objective function values, which were then input to a non-dominated sorting genetic algorithm (NSGA-II) for optimization. In each 
following generation, NSGA-II provided a new set of units for evaluation. The procedure was applied to the single leg of the Chebyshev lambda 
mechanism to better illustrate it, enabling a comprehensive analysis of candidates. Four objective functions (i.e., length in the x-direction, 
trajectory height variation, maximum foot acceleration, and foot speed fluctuation) were used to carry out a multi-objective optimization. The 
calculation time was approximately 2 s/unit.
Keywords: Chebyshev lambda mechanism, optimization, synthesis, walking mechanism
Highlights
• 	 A procedure for the optimization of walking mechanisms was presented.
• 	 Simscape mechanism model was embedded within the NSGA-II; approx. calculation time was 2 s per unit
• 	 Four objective functions were used to optimize the Chebyshev lambda mechanism.
• 	 The proposed procedure can be applied to other types of walking mechanisms.
0  INTRODUCTION
Walking mechanisms imitate the leg motion of walking 
creatures and are comprised of linkages and joints. 
Despite their shortcomings compared to wheeled 
drives, including the lower efficiency, stride height 
variations, and higher complexity, they are useful in 
special applications [1]. Walking mechanisms can 
be used while traversing uneven or stepped surfaces 
[2] where wheeled mechanisms cannot be used. 
Also, they are applicable in regular flat surfaces with 
obstacles [3], e.g., a warehouse with sensitive cables 
on the floor or propelling devices through water via 
paddles [4]. Many types of walking mechanisms are 
available, each with a specific trajectory and number 
of joints and linkages. Thus, the mechanism type 
can be selected depending on the desired outputs. 
Examples include the Chebyshev lambda, Klann, and 
Jansen mechanisms [5] to [7].
The above-listed mechanisms can be applied in 
multiple ways, either using the default linkage values 
or optimizing them for a specific purpose. Komoda 
and Wagatsuma [5] compared the performances 
of the Chebyshev, Klann, and Jansen mechanisms 
regarding energy consumption and trajectory. It was 
found that the simplest of the three (Chebyshev) 
yielded the roughest trajectory, while the most 
complex one (Jansen) resulted in a finer trajectory 
and had higher efficiency. The authors concluded that 
walking mechanisms behave similarly to biological 
evolution based on such results. Moreover, Kim et 
al. [8] compared the performances of three walking 
mechanisms (i.e., four-bar, Klann, and Watt-I) using 
water running as a case. Following the kinematic 
analysis, the former two yielded higher propulsion. 
Additionally, the same mechanism might have 
multiple purposes, given that it is reconfigurable, 
i.e., that linkage lengths can be varied during the 
operation. Sheba et al. [9] presented the reconfigurable 
Klann mechanism to generate various gait cycles: 
digitigrade locomotion, jam avoidance, step climbing, 
hammering, and digging. Foot trajectories were 
obtained analytically, and link dimensions and the 
total number of links were varied. Similarly, Nansai et 
al. [6] and [10] studied and designed a reconfigurable 
Jansen linkage to produce several gait types: 
digitigrade locomotion, obstacle avoidance, jam 
avoidance, step climbing, and drilling motion. Hence, 
it can be concluded that many mechanism types are 
used, while the linkage lengths are often varied, 
especially in reconfigurable mechanisms.
For this reason, the need to adapt the existing 
walking mechanisms to specialized engineering 
problems is evident. This problem can be easily solved 
via optimization, which will, in addition to ensuring a 
feasible solution, yield additional benefits for regular 
and reconfigurable mechanisms. Several studies 
were carried out on the optimization of walking 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
726 Miler ,	D.	–	Bir t,	D.–	Ho ić,	M.
mechanisms, especially when developing novel types. 
For example, Desai et al. [11] presented an eight-link 
walking mechanism of the planar Peaucellier-Lipkin 
type. Once the mechanism geometry was outlined, the 
authors used the genetic algorithm to find the optimal 
linkage lengths. Variation in the foot path was used as 
the sole objective function and was minimized, while 
the transmission angle and stride height were limited. 
The results were verified experimentally. 
Similarly, Erkaya [12] used the genetic algorithm 
to find the Jansen mechanism linkage lengths that will 
result in the minimum foot trajectory variations. While 
developing a legged closed-chain passive-locomotion 
platform, Wei et al. [13] optimized the foot trajectory. 
Two objectives were introduced, vertical variation 
and the longitudinal force between foot and ground. 
The multi-objective optimization was carried out via a 
weighted objective function.
A new mechanism optimization process simplifies 
the mechanism design and improves its outcomes. The 
non-dominated sorting genetic algorithm (NSGA-II) 
was used for optimization, while objective functions 
were evaluated through Simscape Multibody (Matlab 
subroutine). The mechanism’s multi-objective 
optimization was carried out for three objective 
function pairs. Furthermore, boundary conditions 
were included based on Grashof’s law to ensure 
feasibility. The utility of the optimization process was 
exhibited using the Chebyshev’s lambda mechanism 
as an example. Its performance concerning the 
trajectory length (in x-direction), height variation, foot 
speed flux, and foot acceleration was assessed. 
The primary scientific contribution of this paper 
is the optimization process; the numerical Simulink 
calculations were used to determine objective 
function values, which were then input to NSGA-
II for optimization. In each following generation, 
NSGA-II provides a new set of units for the Simulink 
evaluation. The paper is outlined as follows: the 
problem and the associated optimization process were 
outlined in Section 2, along with design variables, 
objective functions, and boundary conditions. 
Algorithm settings were also provided. Further, the 
Simscape model of the mechanism was shown in 
Section 3. Next, the optimization process results were 
presented and discussed in Section 4, while Section 5 
provided key findings and limitations of the presented 
work and a future outlook.
1  METHODS
The Chebyshev lambda mechanism is a simple four-
bar linkage mechanism that converts the rotational 
motion into straight motion. Due to its low trajectory 
height variability, it is a suitable solution for walking 
mechanisms. In this paper, it was studied through 
optimization via the genetic algorithm and Simulink 
Multibody. Due to its symmetry, one-half of the 
mechanism was observed in this paper (see Fig. 1).
The multi-objective optimization of the 
mechanism was carried out using the genetic 
algorithm; the NSGA-II was used to find the 
optimal results (for more details, see Section 2.1). 
Four objective functions were used: length in 
the x-direction, trajectory height variation, speed 
fluctuation, and maximum acceleration. The latter 
two were observed for the output point of the 
mechanism (i.e., its foot). However, due to the 
complexity of analytical expressions and to obtain 
more comprehensive results, Simscape Multibody 
was used to obtain the values of objective functions 
numerically. Hence, objective functions and the 
associated Simscape calculation procedures were 
outlined in Section 2.2. The optimization process 
overview is shown in Fig. 2.
Fig. 1.  Chebyshev’s Lambda mechanism – annotations
The mechanism studied in this paper consists 
of five linkages, with linkages denoted L
1
 to L
5
 with 
default values of {100, 250, 250, 250, 1000} mm. The 
distance between the anchors was L
0
 (see Fig. 1), with 
a default value of 200 mm. Linkages L
1
 and L
2
 were 
anchored, and L
1
 served as the mechanism actuator. 
It should be added that both anchors are located at 
the same height. The sum of default linkage lengths  
L
SUM
 = 2050 mm was used as an input parameter, 
while the lengths were varied (including L
0
). Finally, 
the design variable vector was written as:
 x  LLLLL
01234
    , (1)
while L
5 
was calculated by subtracting the design 
variable vector sum from the L
SUM
. The values of 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
727 Multi-Objective Optimization of the Chebyshev Lambda Mechanism 
design variables L
0
, L
1
, L
2
, and L
4
 were varied with 
respect to the default values used for this type of 
mechanism. Ranges were set to [0.9…1.1] of default 
values. This approach was not applied to L
3
, as it was 
necessary to ensure that selected lengths would result 
in a mechanism that can be assembled for each unit 
within the population (i.e., mechanism feasibility). 
Thus, Grashof’s criterion was used to find the L
3
 
boundary: 
 LLLL
30 12
 . (2)
2.1  Optimization Algorithm Properties
Optimal solutions were found using the NSGA-
II [14]. The NSGA-II is a non-dominated sorting 
genetic algorithm extensively used in engineering 
due to its ensured convergence and absence of 
sharing parameters [15]. As such, it was widely 
applied in multiple engineering fields, with 
examples including materials design [16], design of 
mechanical components [17], and energy consumption 
management [18].
The crossover and mutation rates were defined 
through the crossover distribution index and the 
mutation distribution index. Default values provided 
by the algorithm authors were used; 20 for the 
crossover and 100 for the mutation distribution index. 
It was also necessary to select the population size and 
the number of generations. The selected population 
size of 100 units was improved through 50 generations 
to retain reasonable computational time. Moreover, to 
ensure that the results are near the global optimal, the 
optimization procedure was verified by repeating the 
process using the significantly larger population size 
and the number of generations. The first verification 
was carried out by increasing the number of 
generations to 500 while keeping the population size 
constant. The second verification was the opposite; 
a large population size (1000 units) was used while 
the number of generations was kept constant. Finally, 
algorithm properties were the same for each of the 
observed objective function sets.
2.2  Calculating the Objective Function Values 
As noted earlier, this study considered four objective 
functions: the length in the x-direction, trajectory 
height variation, foot speed fluctuation, and maximum 
foot acceleration. As shown in Fig. 1, the genetic 
algorithm exported the linkage lengths for each of the 
units in the population, which were then automatically 
imported into Simscape. 
The Simscape mechanism model was developed 
as follows: the Chebyshev lambda mechanism 
consisting of linkages, joints, and anchors was 
modelled. Linkage properties were defined, 
including their cross-section and material properties; 
additionally, linkages were connected using pins, 
which were also fully defined. The relations between 
the linkages were achieved using joints, for which 
it was necessary to define degrees of freedom. 
Measurements were carried out at point 5 (mechanism 
foot), located on the free end of the linkage L
5
 (Fig. 2). 
Its position, velocity, and acceleration were output for 
Fig. 2.  Optimization process overview

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
728 Miler ,	D.	–	Bir t,	D.–	Ho ić,	M.
each position during the cycle. The top-level schema 
of the Simscape model is shown in Fig. 3.
Simscape Multibody simulation was carried 
out next and provided all the necessary information; 
foot positions, velocities, and accelerations were 
calculated for the mechanism cycle. The resulting 
points were then used to obtain the values of objective 
functions. The stride cycle was discretized into 3024 
points, and the leg foot positions and accelerations 
were calculated for each point. Hence, the calculation 
step was 0.12°.
2.2.1  Length in the x-Direction
Length in the x-direction (also denoted as Length 
∆x) was selected as the base objective and was thus 
considered in each of the optimization processes. 
It should be noted that it is possible to increase the 
device speed by maximizing the x-direction length 
(given the constant input rotational speed). It was 
calculated based on the leg foot trajectories obtained 
through Simscape simulation. Once the trajectories 
were exported, their minimum and maximum values 
in the x-axis were found and subtracted. Hence, the 
first objective function f1(x) was obtained as:
 fx xx
1
x    
manm in
. (3)
2.2.2  Trajectory Height Variation
Trajectory height variation was used as the second 
objective function and was minimized. Firstly, it 
Fig. 3.  Simscape simulation schematic

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
729 Multi-Objective Optimization of the Chebyshev Lambda Mechanism 
was necessary to determine the points outlining the 
maximum length in the x-direction – M
1
(x
min
, y
M1
) 
and M
2
(x
max
, y
M2
). The trajectory height variation 
denoted as H
var
 was minimized between M
1
 and M
2
 to 
reduce the changes in the vertical direction (Fig. 4). It 
was necessary to reduce the change in height between 
points M
1
 and M
2
 (i.e., to reduce |y
M1
 – y
M2
|), as well 
as to reduce the changes in the x-direction length.
Thus, the objective function was created; the 
difference between y
M1
 and y
M2
 was squared and 
multiplied by the sum of the distances between 
points P
1
, P
2
, P
3
, and P
4
, and the foot trajectory. The 
difference was squared to emphasize the role of the 
difference between heights of points M
1
 and M
2
 while 
also ensuring that it will always be positive. Hence, 
in combination with the distances between the points 
and the foot trajectory, which are always positive, 
trajectory height variation will always be positive. 
Finally, trajectory height variation was expressed as 
the objective function f
2
(x) and minimized:
fH yy
aaaa
2
2
1234
x    
 
varM 1M 2
min( )m in()min( )m in() , (4)
where n denotes the number of discrete points within 
the interval.
The height variation function presented above 
has no physical meaning; it is merely a function 
providing the lowest variations in trajectory height. 
While the difference between the overall maximum 
and minimum y-axis position is simpler to calculate 
and can also be used, it would not guarantee steady 
movement. In this case, the resulting curves had low 
quality and did not result in an industrially viable 
mechanism.
2.2.3  Speed Fluctuation
The speed fluctuation was selected as the third 
objective function. Minimizing the foot speed 
fluctuation will result in reduced accelerations 
throughout the cycle. This will, in turn, reduce inertial 
forces during the cycle, resulting in a smoother 
operation [19]. In this study, the foot speed fluctuation 
was taken as:
 f
vv
n
i
n
3
1
x  


 ia vg
, (5)
where n is the number of discrete points observed 
during one cycle (in this article, n = 3024), v
i
 is the 
resultant speed in each of the points, and v
avg
 is the 
average speed of all the points.
2.2.4  Maximum Foot Acceleration
While reducing the speed fluctuation will result in a 
more uniform acceleration during the stride, it will 
not directly affect its maximum value. However, 
reducing the maximum foot acceleration will result 
in a lower maximum inertial force, often used as 
an input while sizing the drive mechanism. Hence, 
foot acceleration was used as the fourth objective 
function. The maximum acceleration was obtained 
from the Simscape data as a result of x-axis and y-axis 
accelerations and was minimized:
 fa
4
x  
max
. (6)
3  RESULTS AND DISCUSSION
The optimization processes were carried out for three 
sets of objective functions according to the method 
outlined in Section 2. Objective function sets were as 
follows: length in the x-direction and trajectory height 
variation (set 1), length in the x-direction and speed 
fluctuation (set 2), and length in the x-direction and 
acceleration (set 3). Furthermore, it should be noted 
that optimization was carried out using a personal 
computer with an i7-4810MQ processor (2.86 GHz) 
Fig. 4.  The trajectory height variation minimization

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
730 Miler ,	D.	–	Bir t,	D.–	Ho ić,	M.
and 32 GB of RAM. The time needed to evaluate units 
from each set was similar, regardless of the objective 
function, meaning that the Simscape simulation had 
a significantly higher computational cost compared 
to NSGA-II operations. The average calculation time 
was approx. 2 s per unit. Additionally, validation was 
carried out using set 1 as an example (for more details 
on the validation, see Section 2.1).
3.1  Length ∆x and Trajectory Height Variation
The optimization process was first carried out using 
the length in the x-direction and trajectory height 
variation as objective functions. Four Pareto optimal 
solutions were obtained, as shown in Fig. 5. The 
solutions were labelled as “Height” and index ranging 
between 1 and 4, with lower indices representing larger 
strides (and higher variations in trajectory height). 
Hence, the highest x-direction length was obtained for 
Height 1, 584.2 mm, with a height variation value of 
955.6. The lowest trajectory variation was obtained 
for the solution labelled Height 4, 0.01833, with the 
corresponding ∆x = 483.6 mm.
Compared to other optimization processes (see 
Sections 3.2. and 3.3.), only four solutions were found 
on the Pareto front; all remaining final population 
solutions were dominated by the four solutions 
presented in Fig. 5. and Table 1.
Fig. 5.  The x-direction length and height variation
The range between the extreme Pareto solutions 
for ∆x was 100.6 mm (17.22 % of the highest obtained 
value). Solution height variations could not be 
compared in a similar manner as the change was not 
linear (see fitness function f
2
). Thus, to better illustrate 
changes in the height variation and the importance of 
measurement, strides were plotted for all the Pareto 
optimal solutions (Fig. 6).
Table 1.  Set 1 solutions – values
No. L
0
L
1
L
2
L
3
L
4
L
5
1 180.8 108.4 237.7 234.3 272.2 1197
2 189.3 98.67 269.4 270.2 261.5 1150
3 187.9 98.97 235 232.2 252.2 1232
4 186 101.5 231.8 231 237.1 1248
Linkage lengths measured in mm
As shown in Fig. 6, paths obtained for solutions 
Height 1 and Height 2, with respective height variation 
values of 955.6 and 12.03, resulted in a curved bottom 
segment of the leg trajectory. Since the bottom 
segment corresponds with the period during which the 
foot is in contact with the surface, both solutions will 
require larger torque to operate. Changes in height will 
require the driver to lift the device body and provide 
movement in the horizontal direction. In contrast, the 
bottom trajectory segments of solutions Height 3 and 
Height 4, with height variations of 0.118 and 0.018, 
respectively, were remarkably straight. While the 
variations were slightly more prominent in the Height 
3 trajectory, as shown in Fig. 6, it can be concluded 
that both solutions will not require an increase in 
required input torque. Moreover, it can be concluded 
that the height variation of 0.12 is acceptable, but the 
value is strictly limited to the optimization of similarly 
sized mechanisms.
Fig. 6.  Foot paths of Pareto optimal solutions
Finally, none of the variables constituting Pareto 
optimal solutions were in the vicinity of the design 
variable range boundaries. Similarly, the same design 
variables were also not restricted by Grashoff’s 
criterion. Therefore, it was concluded that the selected 
design variable ranges were appropriate for optimizing 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
731 Multi-Objective Optimization of the Chebyshev Lambda Mechanism 
the Chebyshev Lambda mechanism considering the 
∆x length and height variation.
3.2  Change in the x-direction and Speed Fluctuation
The optimization process was carried out for the ∆x 
length and speed fluctuation along the stride. Solutions 
on the Pareto front and the associated linkage lengths 
are provided in Fig. 7 and Table 2. Similarly to Section 
3.1, Pareto optimal solutions were indexed from 1 to 
11, starting with the solution having the longest ∆x 
length. Additionally, the label “Speed” was included 
before the index. The highest obtained ∆x was 487.3 
mm; the same solution yielded a speed fluctuation of 
7426 (Speed 1). On the other side of the Pareto front, 
the lowest speed fluctuation of 6861 was obtained, for 
which ∆x = 464.7 mm (Speed 11).
Fig. 8.  Length	∆x and maximum acceleration
Table 3.  Set 3 solutions – values
No. L
0
L
1
L
2
L
3
L
4
L
5
1 216 90.63 250.5 157.2 265.5 1286
2 215.8 90.64 251.1 158.1 263.1 1287
3 216.5 90.67 249.8 156.1 263.4 1290
4 216.3 91.12 249.3 159.2 263.6 1287
5 215.8 90.80 251.5 160.2 261.9 1286
6 215.8 90.83 251.4 160.7 261.9 1285
7 216.4 90.58 249.1 156.8 263.3 1290
8 216 90.15 251.0 159.5 261.6 1288
9 216.5 90.20 248 156.4 262.6 1298
10 216.6 90.02 247.8 156.1 262.1 1294
11 217.3 90.13 248.8 156.8 262.1 1292
Linkage lengths measured in mm
Fig. 7.  The x-direction length and speed fluctuation
Table 2.  Set 2 solutions – values
No. L
0
L
1
L
2
L
3
L
4
L
5
1 206.4 97.18 270.2 172.5 241.5 1268
2 206.6 97.37 272.0 177.0 240.8 1263
3 206.6 97.13 269.5 172.6 240.3 1271
4 207.3 96.44 268.6 169.6 240.0 1275
5 191.0 90.17 252.6 160.8 234.9 1312
6 191.2 90.38 252.5 162.2 235.4 1310
7 189.6 90.16 252.5 165.0 235.0 1307
8 191.7 90.12 251.0 159.9 235.1 1314
9 191.4 90.09 251.0 160.3 235.1 1314
10 218.6 97.95 270.2 161.3 237.8 1283
11 217.2 97.51 271.2 164.8 238.1 1278
Linkage lengths measured in mm
3.3  Change in the x-direction and Acceleration
The x-direction length and maximum foot acceleration 
were selected as the third pair of objective functions. 
This combination was to provide a mechanism that 
will result in a steady and fast movement, and the 
results are presented in Fig. 8 and Table 3. Eleven 
Pareto optimal solutions were found, labelled as 
“Acceleration” 1 to 11, with lower indices associated 
with larger ∆x values and higher accelerations; the 
opposite was true for higher indices. The largest ∆x 
length of 445.1 mm was obtained for the solution 
Acceleration 1, corresponding to the highest 
acceleration between the solutions (122.7 m/s
2
). In 
contrast, the lowest acceleration of 117.3 m/s
2
 was 
obtained for Acceleration 11 (∆x = 432.7 mm). The 
ranges between the extreme Pareto solutions were 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
732 Miler ,	D.	–	Bir t,	D.–	Ho ić,	M.
speed fluctuation was 4.32e
4
. Finally, the maximum 
acceleration of the default mechanism foot was 2.27e
5
 
m/s
2
.
The default value was compared to designated 
solutions from the Pareto optimal fronts for each 
pair of objective functions, as shown in Fig. 9. The 
objective function outputs for each of the solutions in 
the diagram were divided with the corresponding value 
obtained for the default solution. The only exception 
was the trajectory height variation, normalized to a [0, 
1] interval via min-max normalization supplemented 
with a logarithmic operator. This was necessary to 
retain clarity, as the objective function outputs for 
trajectory height variation were exponential. They 
were normalized as follows:
 H
HH
HH
i
i
'
logm in log
maxl og minl og
.
var
varv ar
varv ar

 
  
 (7)
As shown in the figure, solutions obtained 
while optimizing for length ∆x and trajectory height 
variation (Height 1 and 4) yielded larger values 
than the default solution. While Height 1 ∆x was 
significantly larger compared to the default solution, 
so was its height variation. In contrast, Height 4 ∆x 
was 13.2 mm longer, having lower height variation, 
speed fluctuation, and slightly higher maximum 
acceleration. Compared to other solutions (barring 
Height 4), the default one had a significantly lower 
trajectory height variation. Thus, it is recommended to 
include the height variation as a boundary condition 
when designing a walking mechanism.
Moreover, when considering the speed fluctuation 
criterion, the default solution resulted in the second-
highest value (behind the Height 1 solution). Solutions 
12.34 mm for the ∆x length (2.77 % of the highest 
obtained value) and 4.4 m/s
2
 for the acceleration 
(4.4 %). Thus, it can be concluded that changes in 
acceleration were more prominent when observing the 
Pareto front.
Regarding the variable boundary ranges, the only 
active condition was for the lower L
1
 value, which 
was set to 90 mm. All the Pareto-optimal solutions 
stemmed toward the lower variable boundary, 
implying that when aiming to increase the ∆x length 
and decrease the foot acceleration, the L
1
 range should 
be increased further. The distance between the anchors 
L
0
 in optimal solutions ranged between 215.8 mm and 
217.3 mm, illustrating a rather low variance. Linkage 
L
2
 length was practically equal to the default value, 
while the lengths of linkages L
3
 differed the most with 
respect to the defaults.
Obtained accelerations are theoretical and much 
higher than their experimental counterparts. This is 
due to the fixed rotational velocity at the mechanism 
input. The constant rotational velocity implies infinite 
torque; in other words, realistic loads stemming from 
the mechanism operation will not slow down the 
motor. Hence, high acceleration values were obtained, 
resulting in significant inertial forces within the 
system. In a realistic case, if operated via motor, such 
large accelerations would not occur as the increased 
inertial forces would slow down the motor.
3.4  Comparison to the Default Mechanism
Furthermore, to compare previously obtained Pareto 
fronts to the default mechanism, its characteristics 
were determined next. The default mechanism 
x-direction length was 470.4 mm. A trajectory height 
variation of 0.2799 was also obtained, while the 
Fig. 9.  Comparison between the default and optimal solutions

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 725-734
733 Multi-Objective Optimization of the Chebyshev Lambda Mechanism 
obtained when minimizing the acceleration had a low-
speed fluctuation, as expected. Similarly, solutions 
obtained when aiming to reduce the speed fluctuation 
(Speed 1 and Speed 11, for more details, see Section 
3.2.) resulted in low maximum acceleration values. 
Such behaviour was expected, as speed fluctuation 
and maximum acceleration are linked indirectly.
Finally, it should be clarified that in Fig. 9, 
a higher ∆x length is considered positive, while 
higher speed fluctuation, maximum acceleration, 
and trajectory height variation values imply a lower-
quality solution. Moreover, in the trajectory height 
variation graph, the solution Height 4 column is 
missing; it had the lowest value of all the seven 
observed solutions and was normalized to 0.
4  CONCLUSIONS
The procedure for multi-objective optimization 
of walking mechanisms based on the Simscape 
Simulation embedded within the NSGA-II was 
presented. The simplest of the mechanisms, the 
Chebyshev Lambda mechanism, was used as an 
example and was optimized. The proposed method 
also applies to both 6-bar and 8-bar linkages, with 
slight differences in integration. A new computer-
aided design (CAD) model should be created, 
and corresponding design variables and boundary 
conditions should be selected.
The variable ranges were set based on the 
default mechanism, and Grashoff’s criterion was 
used to ensure that the resultant solutions would be 
viable. Optimal solutions were found for three pairs 
of objective functions: ∆x length/height variation, ∆x 
length/speed fluctuation, and ∆x length/maximum 
foot acceleration. Based on the presented results, the 
following conclusions were made:
• The presented procedure enabled a timely 
and comprehensive analysis of the candidate 
mechanisms; the calculation time was 
approximately 2 s per unit.
• Considering the length in the x-direction and 
height variation, notable advancements were 
made. For the Pareto optimal solution with the 
highest length, the increase with respect to the 
default solution was 24.2 %. Moreover, all the 
Pareto optimal solutions had higher ∆x lengths 
than the default solution.
• It is recommended to introduce the trajectory 
height variation as a boundary condition when 
aiming to use the Chebyshev Lambda mechanism 
as a walking mechanism.
However, the limitations of the presented study 
should be addressed as well. The proposed method 
considers the walking mechanism without taking into 
account its surroundings. A device utilizing walking 
mechanisms for movement will include multiple 
such mechanisms; hence, influences of operating 
conditions and gaits (e.g., biped, quadruped, hexapod) 
should also be considered. Furthermore, when using 
the results of this study, it is necessary to acknowledge 
that a minimal number of boundary conditions was 
used as the authors primarily observed the effects of 
objective functions on the solution characteristics. 
Hence, some of the Pareto solutions would result in 
sub-optimal walking mechanisms due to variations in 
length in the x-direction. 
For this reason, in the future study, we plan 
to generate a mechanism through multi-objective 
optimization by simultaneously introducing all the 
objective functions (instead of pairs, which were 
considered here). Finally, it should be added that this 
study only considered the mechanism kinematics; 
hence, the viability of the optimal mechanism should 
be verified by either simulating its dynamics or 
experimentally.
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