Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100 Received for review: 2020-10-12
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-01-29
DOI:10.5545/sv-jme.2020.6990 Original Scientific Paper Accepted for publication: 2021-02-05
*Corr. Author’s Address: Universidad Autónoma de Ciudad Juárez, Av. del Charro 450, Ciudad Juárez, Chihuahua, México, jose.davalos@uacj.mx 88
Optimization of Running Blade Prosthetics Utilizing  
Crow Search Algorithm Assisted by Artificial Neural Networks
Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
Manuel Javier Rosel Solis
1
 – José Omar Dávalos Ramírez
2,*
 – Javier Molina Salazar
2
 –  
Juan Antonio Ruiz Ochoa
1
 – Antonio Gómez Roa
1
1
Autonomous University of Baja California, México 
2
Autonomous University of Ciudad Juárez, México
A crow search algorithm (CSA) was applied to perform the optimization of a running blade prosthetics (RBP) made of composite materials 
like carbon fibre layers and cores of acrylonitrile butadiene styrene (ABS). Optimization aims to increase the RBP displacement limited by 
the Tsai-Wu failure criterion. Both displacement and the Tsai-Wu criterion are predicted using artificial neural networks (ANN) trained with 
a database constructed from finite element method (FEM) simulations. Three different cases are optimized varying the carbon fibre layers 
orientations: –45°/45°, 0°/90°, and a case with the two-fibre layer orientations intercalated. Five geometric parameters and a number 
of carbon fibre layers are selected as design parameters. A sensitivity analysis is performed using the Garzon equation. The best balance 
between displacement and failure criterion was found with fibre layers oriented at 0°/90°. The optimal candidate with –45°/45° orientation 
presents higher displacement; however, the Tsai-Wu criterion was less than 0.5 and not suitable for RBP design. The case with intercalated 
fibres presented a minimal displacement being the stiffer RBP design. The damage concentrates mostly in the zone that contacts the ground. 
The sensitivity study found that the number of layers and width were the most important design parameters.  
Keywords: optimization, crow search algorithm, artificial neural networks, running blade prosthetics, Tsai-Wu criterion, finite element 
method 
Highlights
• 	 An optimization methodology was proposed for the RBP design.
• 	 A new technique for RBP manufacturing was evaluated.
• 	 Carbon fibre layers with layers oriented 0°/90° were the best option for the design of the RBP.
• 	 The proposed methodologies reduce the manufacturing costs of the RBP.
0  INTRODUCTION
A prosthesis is an artificial element integrated into 
the human body to replace an internal or external 
organ. The most common prostheses are those that 
replace upper and lower limbs. With the use of new 
materials and better manufacturing techniques, it 
has been possible to optimize prosthetic models to 
replace amputated arms and legs [1] to [3]. Regarding 
leg prostheses, the applications are diverse; everyday 
cosmetic prostheses have been created that allow 
an amputated person to lead a normal life, thereby 
improving their living conditions. Furthermore, 
advanced prostheses have been developed that have 
allowed an athlete with amputations in one or both 
lower limbs took take part in sports activities almost 
at the level of a non-amputee athlete. An example is 
the blade type prostheses used by the South African 
athlete Oscar Pistorius made with carbon fibre, a 
composite material with applications in the automotive 
and aerospace industry, which has properties similar 
to steel but with little weight, which makes it ideal for 
prosthetic applications. Manufacturers of these sports 
implement, for low and high impact activities, such 
as Ösur and Ottobock, offer various blade-type sports 
prostheses, but the high cost of these makes them 
inaccessible to the general public [4]
An alternative in the manufacture of sports 
prostheses, like running blade prosthetics (RBP), is 
the inclusion of additive manufacturing in the process 
of developing a human prosthesis, in particular, 
fused deposition modelling, which is one of the most 
widely used processes due to its simplicity and ease 
of operation. Therefore, the mechanical properties 
that composite materials offer can be combined 
with the advantages of fused deposition modelling, 
mainly in small-scale production, with the option of 
manufacturing highly personalized products and, at 
the same time, with shorter prototypes development 
times [5]. 
Türk et al. [6] evaluate the mechanical 
performance of lower limb prostheses manufactured 
combining additive manufacturing with carbon 
fibre-reinforced polymers. They found that by using 
this technique, the mechanical strength increases by 
around 40 %, and the weight decreases by 28 %. 
Tavangarian et al. [7] studied the mechanical 
strength of the pylon section of a lower limb prosthetic 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
89 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
manufactured with 3D printing using polylactic acid 
polymers. They found that this material meets the 
standards set by ISO 1038:2016 [8] regarding the 
structural requirements of lower-limb prostheses.  
Ouarhim et al. [9] perform the characterization 
and numerical simulation of laminated glass fibre-
polyester composites applied to running blade 
prosthetics. They used software to simulate the 
composite materials. Its results show that the number 
of layers impacts in bending and buckling. Its results 
show a good agreement between experimental and 
numerical. 
A good balance between displacement 
and mechanical resistance guarantees the good 
performance of an RBP. During the change of length 
or displacement of an RBP, energy is stored and 
returned, increasing the forward propulsion [10]. 
However, increasing the displacement can lead to the 
mechanical failure of RBP, mainly in the sections with 
a curved shape.
To achieve an optimal design of an RBP , advanced 
tools as optimization algorithms are required. These 
types of algorithms are widely used to solve diverse 
problems in engineering fields [11] to [14]. A common 
characteristic of optimization algorithms is the large 
number of objective function evaluations required to 
find an optimal solution. When experimental or finite 
element method (FEM) calculations are employed 
to perform objective function evaluations, the 
computational cost increases, making the optimization 
process infeasible. In these cases, surrogated methods 
are an option to reduce the optimization time [15] 
to [17]. An example of an optimization strategy 
assisted by surrogated methods applied to lower limb 
prosthesis manufactured with composite materials 
can be found in [18], focusing on the optimization 
of the pylon tube made of isogrid structures. Due to 
optimization, they achieved a reduction of 13.3 % 
and 70.59 % in the mass and the Tsai-Wu index under 
compression effects. 
Unlike previous optimization work, the present 
investigation is applied to a lower-limb sport 
prosthetics which can be used to reintegrate people 
with lower-limb amputations into sport activities. 
Also, optimization methodology employs artificial 
neural networks (ANN) with two intentions: to 
evaluate the objective function and to use the ANN 
coefficients in the calculation of relative importance 
of design parameters. 
This work employs an optimization strategy to 
improve the design of an RBP. This strategy integrates 
crow search algorithm (CSA), and ANN and FEM 
calculations. The objective of the optimization was 
to find a design that has better resistance to failure 
without decreasing its capability to displace in order 
to maintain its functionality. The failure criterion used 
for determining the resistance of the RBP was the 
Tsai-Wu criterion. To reduce the computational cost, 
displacements and the Tsai-Wu failure criterion were 
calculated by means of ANN. The ANN was trained 
with a database constructed from FEM calculations of 
displacement and Tsai-Wu failure. The manufacturing 
process, modelled using commercial FEM software, 
is a lay-up process with carbon fibre layers stacked 
around acrylonitrile butadiene styrene (ABS) cores. 
Three carbon fibre orientations were tested to find 
the better design option: –45°/45°, 0°/90°, and a 
combined case with –45°/45° and 0°/90° orientations. 
Using the ANN coefficients, a sensitivity analysis 
was performed to know the relative importance of the 
design parameters in each orientation.
1  METHODS
1.1  RBP Design and Manufacturing
An RBP is a hook-shaped artefact made primarily of 
composite materials from epoxy matrix carbon fibres. 
The manufacturing processes used for its manufacture 
are the lay-up process and the pre-impregnated fibre 
process. The first consists of applying or mixing epoxy 
resin with a brush or roller into the carbon fibre layers 
that are stacked until reaching the thickness based 
on the desired resistance. Another manufacturing 
process consists of the use of pre-impregnated carbon 
fibre layers that, after being stacked, must undergo a 
thermal treatment in an autoclave for the activation of 
the resin [19]. The RBPs have a spring-like mechanical 
behaviour; during the first gait cycle, the prosthesis is 
compressed by the effect of the dynamic load of the 
body storing energy. In the second stage of the gait 
cycle, that stored energy is released, generating a 
thrust forward, just as the heel joint of the human foot 
does. The maximum efficiency in this energy return 
of a high-performance RPB ranges between 63 % and 
95 % of the accumulated energy [20]. 
1.2 Optimization Methodology
A bio-inspired method used in optimization is the 
CSA. This novel algorithm, developed by Askarzadeh 
[21], is based in the behaviour of crows, considered 
the most intelligent birds. A characteristic of the 
crow’s behaviour is the way they store food in 
hideouts to prevent that other crows from stealing it; 
at the same time, they look for opportunities to steal 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
90 Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
the food of other members of the flock. Crows use the 
gained experience stealing food to deceive potential 
plunderers, employing diverse strategies [22]. Crows, 
always try to find optimal food places. This intelligent 
behaviour is replied in the CSA in which the optimal 
place is analogous to the global optima into the search 
space. In CSA, it is assumed that the crows live in 
flocks, can memorize the places where they hide their 
food, a crow follows another to try to steal its food and 
protect its hideouts from other crows via a probability 
reason [22]. 
In the CSA optimization process, during a iteration 
iter, a crow i, from a flock of size N, occupy a position 
defined by the vector xx xx
ii ti it ii t
d
ii t ,, ,,
,, ,  



12
, where 
i = 1, 2, ..., N,  it = 1, 2, ..., it
max
, it
max
 is the maximum 
number of iterations, and d is the dimension of the 
problem or the number of variables in the design. The 
position, m
i,it
, of the hideout it is stored in the crow 
memory and represents the best position at it, the far. 
For the next iterations, the crow will try to find better 
positions.
Two situations may occur when, at iteration it, 
a crow j visits the hideout where it stores its food 
and a crow i is on the lookout attempting to steal it: 
(1) the crow j does not realize that crow i followed 
it and reveals its hideout to the thief crow; (2) the 
crow j detects the presence of the crow i and changes 
its trajectory to avoid revealing the location of its 
hideout. In the first situation, the new position of crow 
i is defined by:
 xx rf lm x
ii ti it
i
ii ti it ii t ,, ,, ,
,

  

1
 (1)
where 0 ≤ r
i
 ≤ 1 is a random number with uniform 
distribution and fl 
i,it
 is the flight length of crow i at 
iteration it. If the second situation arises, the crow j 
will change its trajectory through fake hideouts to 
deceive to crow i. Both situations are expressed as 
follows:
  x
xr fl mx rA P
ii t
it
i
ii tj it ii t
j
ji t
,
,, ,, ,

 


a random positio on otherwise





, (2)
where 0 ≤ r
j
 ≤ 1 is a random number with uniform 
distribution and AP  
j,it
 is the awareness probability 
of the crow j at iteration it. Intensification and 
diversification are two important characteristics of 
metaheuristic algorithms; these characteristics in 
CSA are provided by fl and AP. Lower values of both 
parameters will guide the optimization to local search, 
increasing the intensification, whereas larger values 
expand the search randomly in the decision space 
but reduce the probability of found optimal solutions, 
resulting in increasing diversification.
In Fig. 1, the flowchart of the CSA is presented. 
In the first step, the objective function, the range 
of decision variables and constraints are defined, 
also, the CSA parameters, N, it
max
, d, fl, and AP are 
initialized. In Step 2 the positions of flock crows are 
defined randomly; because the crows do not have 
prior experience at Iteration 1, it is assumed that the 
initial position is the location of the hideout. In Step 
3, the algorithm evaluates the objective function. 
In the next step, all the crows of the flock update its 
position using the next procedure: a crow i randomly 
selects a crow j of the flock and tries to discover the 
position of the hideout, m 
j
. The position is updated 
using Eq. (2). In Step 5, if the new position of crow 
i is feasible, then its position is updated; if not, crow 
i will remain at the initial position. In the next step, 
the objective function is evaluated for the new crow 
flock positions. In Step 7, the memory of the crow 
flocks is updated. A comparison is made between the 
value of the objective function evaluated with both 
the new position x
i,it+1
 and the position m
i,it
. If the new 
position performs better. the memory it is updated; 
if not, the memory remains without changes. This is 
mathematically stated as follow:
 m
xFxF m
m
ii t
ii t
obj
ii t
obj
ii t
ii t
,
,, ,
,

 


 
1
11 1
better than
1 1
otherwise





. (3)
Fig. 1.  CSA flowchart

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
91 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
1.3  Artificial Neural Network
An ANN is a machine-learning technique used in the 
prediction of output variables as a function of input 
variables. A typical ANN consists of processing 
elements or neurons grouped in layers sequentially 
connected by synaptic weights. The most popular type 
of ANN is the multi-layer perceptron (MLP) with at 
least three neurons layers divided into input, hidden, 
and output layers. The number of neurons clustered 
at input and output layers is equal to the number of 
input and output variables, respectively, whereas, in 
the case of the hidden layer, the number of neurons 
is defined in terms of the predictive accuracy level 
of the ANN model [23] to [25]. The output of layers 
is computed by means of transfer functions whose 
argument is the summation of the multiplication of 
the corresponding weight by input signals plus a bias 
[26] a comprehensive review of the artificial neural 
network (ANN. The ANN prediction is computed by 
the next expression:
 ANNT Fw TF b
output
O
j
n
jj
H











1
22
. (4)
Here ANN
output
 is the ANN prediction, TF 
O
 is 
the transfer function of the output layer, w is the 
connection weight, TF 
H
 is the transfer function of the 
hidden layer, b is the bias, n is the number of neurons 
in the hidden layer, and the superscript 2 indicates the 
output layer. 
The ANN is trained using a set of sample data. 
During the training, the ANN coefficients, w and b, 
are proposed and the error between ANN
output
 and real 
outputs, stored in a database, is calculated. This error 
is minimized by adjusting the coefficients by means of 
a training algorithm.
1.4  Tsai-Wu Failure Theory
The Tsai-Wu failure [27] theory is widely used in 
the prediction of failure in laminates. In the case of 
laminate analysis, the governing equation in matrix 
form is:
 
N
M k
AB
BD
























0
. (5)
Here, N and M are the resultant force and moment 
respectively, 



0
0
0
0











x
y
xy
 and k
k
k
k
x
y
xy











 are the 
displacement in the midplane and curvature effects 
respectively, A, B and D are the extensional, coupling 
and bending matrices, which are calculated as follow:
 At t
k
n
kk


 

 
1
1
Q
k
, (6)
 Bt t
k
n
kk





 
1
2
1
2
Q
k
, (7)
 Dt t
k
n
kk





 
1
3
1
3
Q
k
. (8)
Here, n is the total number of layers, Q is the 
transformed stiffness matrix, and t is the thickness of 
the composite layer. Using the stress-strain 
relationship for laminates, it is possible to calculate 
the state of stress of each composite layer, as is stated 
in Eq. (9):
 




x
y
xy
x





















QQQ
QQQ
QQQ
11 12 16
12 22 11
16 26 66
0
 

y
xy
0
0










, (9)
where σ
x
 and σ
y
 are the normal stresses in x and y 
directions respectively, τ
xy
 is the shear stresses and  
γ
xy
0
 is the shear strain. The calculation of the stress 
state allows using the Tsai-Wu failure criterion:
HH HH HH
11 22 61 21 11
2
22 2
2
66 12
2
1     . (10)
The Tsai-Wu criterion states that the failure 
occurs when this condition is not satisfied. In the FEM 
software used, this condition is associated with the SF 
in composite materials so that in FEM simulations, if 
SF < 1 the failure is expected.
2  PROBLEM FORMULATION
This work focuses on obtaining the optimal design 
variables that allow the maximum displacements of 
an RBP constrained by the safety factor (SF) based 
on the Tsai-Wu failure theory. The process used in 
the manufacturing of the RBP is the lay-up process, 
in which layers of 3k woven carbon fibre and epoxy 
resin are used. The fibre layers are placed in –45°/45° 
and 0°/90° orientations. 
The strategy employed consists of obtaining the 
design parameters through an optimization algorithm 
that evaluates the objective function that defines 
the optimization problem; nevertheless, during the 
optimization, it is not feasible to evaluate it by means 
of numerical methods due to excessive consumption 
of time and computational resources. To improve 
the optimization process, a mathematical model is 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
92 Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
constructed using a surrogated method. This model 
will predict both the displacement and SF, which 
are required in objective function evaluations. 
To obtain this model, it is necessary to construct 
a database that contains a set of combinations of 
design parameters with their respective responses, in 
this case, displacement and SF. The responses to be 
obtained from FEM calculations due to the number of 
required simulations is significantly lower than in the 
optimization process. Employing this strategy led to a 
reduction of the required time to design an RBP. For 
this research, the chosen optimization method was the 
CSA, whereas ANN was used as surrogated method.
Fig. 2.  Optimization procedure
The sequence of activities to optimize the 
RBP is described in Fig. 2. First are defined the 
design variables with its bounds and the prothesis 
load conditions. Afterward, the FEM calculations 
are performed to generate the database whose 
combinations are based on a design of experiments in 
order to promote the learning of the ANN. In the next 
step, the database was used to train and test the ANN 
models that will predict the prosthesis displacements 
and the SF, respectively. In this step, the ANN 
coefficients, weights, and biases are obtained. Before 
the optimization, the relative importance of each 
of the design variables in the variables predicted is 
calculated by means of a sensitivity analysis. Finally, 
in the optimization process, the CSA parameters are 
defined, and the objective function is evaluated, 
obtained as a result the optimization of the RBP. 
This process is performed three times for each of the 
fibre layer orientations. A detailed description of this 
strategy is presented in the next sections.
2.1  CSA Parameters
Three optimizations were performed for different 
layer orientations: a first case, C1, with layers of 
carbon fibre with orientation –45°/45°, a case C2 with 
layers of carbon fibre with orientation 0°/90° and 
a third case, C3, with a combination of carbon fibre 
layers oriented –45°/45° and 0/90°. The objective 
function that defines the problem is expressed by:
 min, Fd
obj ANN
 15 (11)
where F
obj
 is the objective function, d
ANN
 is the 
displacement calculated by means of ANN and 15 is 
a maximum displacement, expressed in mm, expected 
during the optimization. This value was set based 
on the results obtained from the central composite 
design of experiments. The objective function must be 
minimized because it is expected that as the obtained 
displacement increases during CSA iterations, the 
value of the function approaches zero. The objective 
function is subjected to:
 12 0 .,  SF
ANN
 (12)
 xx x
i
lb
ii
ub
≤≤ , (13)
here SF
ANN
 is the safety factor predicted using ANN, 
x
i
 is the i
th
 design variable, and the superscripts lb and 
ub are the lower and upper bound, respectively. 
The same optimization parameters were used for 
the three optimization cases. Each optimization was 
carried on during 1000 iterations; the flock size was 
50 crows, fl was 2, and AP was 0.1.
2.2  Parameterization
In Fig. 3, the geometric segments of the RBP profile 
are illustrated. For the optimization, the dimensions of 
two of them stay constant. These two segments belong 
to the joint with the human body (top of the RBP), 
with a length of 97.12 mm, and the contact area with 
the ground (bottom of the RBP), with an arc length of 
10 mm. The remaining four segments, associated with 
the letter R, are defined geometrically by radii whose 
dimensions were selected as design parameters for the 
optimization. To guarantee a smooth transition, the 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
93 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
points of segments R1, R2, R3 and R4 are tangential to 
each other. The last geometric parameter corresponds 
to the width, W, of the RBP. An additional parameter 
is the number of layers of carbon fibre that cover the 
ABS core. This parameter must be an even number to 
maintain the same number of layers on both the front 
and the back of the RBP. The bounds of the design 
parameters were the same for the three optimization 
cases and are listed in Table 1. The geometry used 
was a surface modelled in CAD software, based on 
the dimensions of a commercial prosthesis [28]. The 
geometry profile was obtained from the centreline 
of the original one, as it was required to generate the 
surface necessary for the simulation of a FEM module 
for composite materials.
Fig. 3.  Geometrical parameters of the RBP
Table 1.  Design parameter bounds
Design parameter Lower bound (lb) Upper bound (ub)
R1 [mm] 106.2 129.8
R2 [mm] 74.7 91.3
R3 [mm] 283.5 343.5
R4 [mm] 110.7 135.3
W [mm] 40.5 49.5
No. of layers [-] 18 26
2.3  Generation of Database and FEM Simulations
For each optimization case, a database based 
on a central composite design of experiments was 
generated to be used in the ANN predictions. The size 
of each database was 6×125, where 6 is the number 
of input variables or design variables and 125 is the 
number of design combinations in the database. The 
dimensions of each combination varied according 
to the design of experiments, except for those 
remining constant, as was referred in Fig. 3.  For 
each combination, the displacement and the SF using 
static linear FEM calculations were calculated. To 
characterize the materials in the simulations, a special 
module for composite materials was used. Here, 
were set the inputs shown in Table 2; the number of 
layers, in even number, to maintain the symmetry of 
the prosthesis, the materials that make up the ABS 
composite material, fibre, core, and combinations of 
fibre orientations. Properties of carbon fibre and ABS 
are presented in Tables 3 and 4, respectively.
Table 2.  Input parameters for composite materials
Fabrics ABS, Core
Number of layers 18, 20, 22, 24, 26
Sub laminates –45°/45 °, 0°/90°
Table 3.  Properties of 3k woven carbon fibres
Properties Unit Value
E
x
[MPa] 59160
E
y
[MPa] 59160
E
z
[MPa] 7500
υ
xy
[-] 0.04
υ
yz
[-] 0.3
υ
xz
[-] 0.3
ρ [kg/m
3
] 1451
Table 4.  Properties of ABS
Property Unit Value
E [MPa] 2588
υ [-] 0.36
ρ [kg/m³] 1040
This configuration was exported to a static 
structural analysis, where each layer was discretized 
using shell elements, which have 4 nodes per element 
and 6 degrees of freedom per node. The finite elements 
have a size of 2 mm resulting in a mesh size of 6400 
elements per layer (Fig. 4a). The number of elements 
was selected based on the comparison between FEM 
and experimental measurements.
The load was applied in the top segment of the 
RBP, whereas the displacement was constrained 
in all directions in the segment of contact with the 
ground (Fig. 4b). The magnitude of the load was set 
considering a human weight of 77.6 kg, which results 
in a static load of 761.25 N. For dynamic conditions, 
the load was increased 2.7 times, which corresponds 
to a maximum load peak during running [29]. Then, 
the applied load was 2055 N.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
94 Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
Fig. 4.  a) Discretized model, and b) force and constraints 
From the FEM simulations, the displacement of 
RBP and the SF based on Tsai-Wu failure theory were 
obtained. 
2.4  ANN Modelling
To reduce the computational time during the design 
optimization, a total of six ANN models were trained 
to predict the displacement and SF factor for each of 
the optimization cases. The ANN was trained using 
the Levenberg-Marquardt algorithm [30] and [31] 
with a learning rate of 1×10
–4
. The transfer function 
for the hidden layer, TF 
H
, was the hyperbolic transfer 
function Eq. (14), whose argument is defined by Eq. 
(15), whereas output layer TF 
O
 was linear [32] and 
[33].
 TF
e
j
H
x
j




2
1
1
2
, (14)
 xw in b
j
i
m
ii n





1
11
, (15)
Here x
j
 is the argument for the transfer function 
of the j
th
 hidden neuron, m is the number of neurons in 
the input layer or number of inputs and the superscript 
1 indicates the hidden layer.
The architecture of the net consists of an input 
layer with six neurons and one neuron at the output 
layer. In the hidden layer, all the nets that were used to 
predict the displacement and Tsai-Wu criterion in the 
optimization cases contain two neurons. 
Eighty per cent of data samples were used for 
training, and the rest for testing. Input data were 
normalized between 0.1 and 0.9 by means of [34]:
 in
In In
In In
i
il ow
upp low











 08 01 .. , (16)
where in
i
 is the normalized input, In
i
 is the 
unnormalized input, In
low
 is the input lower bound and 
In
upp
 is the input upper bound. 
During training and testing, the ANN was 
evaluated using the mean squared error, and the 
correlation coefficient defined by Eqs. (17) and (18), 
respectively:
 RMSE
yy
T
t
T
ANN FEM

 


1
2
, (17)
 R
yy
yy
t
T
ANN FEM
t
T
FEM ave
2 1
2
1
1 
 
 




, (18)
where y
ANN
 is the output predicted by ANN, y
FEM
 is 
the output predicted by FEM, y
ave
 is the average of 
actual values and T is the number of samples. 
2.5  Sensitivity Analysis
A sensitivity analysis enables determining the 
relative importance of each of the design variables 
in the displacement and in the Tsai-Wu criterion. 
For this purpose, the equation proposed by Garson 
[35] was used. Eq. (19) is suitable due its use of the 
ANN synaptic weights to calculate the percentage of 
importance of each variable. In addition to relative 
importance, the obtained results of the sensitivity 
analysis can provide information about of the relation 
between the design variables. 
 I
w
w
w
w
w
m
j
n mn
i
m
in
jm
i
m
j
n in
i
m
in



















1
1
1
1
2
11
1
1
1
w w
jm
2








, (19)
3  RESULTS
3.1  Numerical Validation
A comparison was made between FEM simulations 
and experimental test results reported by Rosel-
Solis et al. [36]. The specimen was manufactured by 
the lay-up process with carbon fibres with 0°/90° 
orientation. The machine was a Shimadzu AG-IC 
stress floor machine with a load cell of 100 kN. The 
material conditions were the same as described above, 
whereas the force and support conditions were set as 
in the experiment. The comparison is shown in Fig. 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
95 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
5. The maximum difference between experimental 
and numerical displacements was around 8 % caused 
by a force near 600 N. From there, the difference 
decreases considerably. Based on these results, it can 
be stated that the proposed methodology to calculate 
displacements and SF is highly suitable for this 
purpose.
3.2  ANN Results
Six models were trained to vary the number of hidden 
neurons to obtain the best ANN performance. The 
architecture obtained was 6:2:1 for all ANN models. 
This means that two neurons are in the hidden 
layer. The comparison of FEM and ANN results 
for displacement and SF are presented in Fig. 6. A 
good approach is observed in the prediction of both 
variables. The values of R
2
 were > 0.99 for all models, 
whereas the values obtained of MSE were < 1.3×10
-–2
. 
Fig. 5.  Comparison between experimental and numerical results
Five new FEM simulations, with different design 
parameters than the stored in database, were performed 
Fig. 6.  Comparison of ANN and FEM predictions; a) C1 displacement, b) C1 SF, c) C2 displacement, d) C2 SF, e) C3 displacement, f) C3 SF 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
96 Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
to test each ANN models. The results are presented in 
Fig. 7. The prediction of displacement and SF in C1 
had an average error of 5 % and 2 %, respectively. 
For C2, the average error for displacement was 3.5 
and for SF was 3.2 %, whereas in C3 the ANN model 
predicted the displacement and SF with an average 
error of 4.1 % and 4.2 %, respectively. These findings 
guarantee the capability of ANN accurate predictions 
with the proposed architecture.
Fig. 7.  Testing of ANN models 
3.3  Optimization Results
The optimization convergence for three cases is 
presented in Fig. 8. Around the first 300 iterations, 
the CSA has identified the region of the search space 
where the optimal solutions are located. Minimal 
changes are observed in the objective function value 
from the subsequent iterations.
Fig. 8.  CSA optimization convergence for:  
a) C1, b) C2, and c) C3
The optimized results of displacement and SF 
are presented in Table 5. In C1, with fibres oriented 
–45°/45°, no solutions that meet the requirements of 
an SF > 1.2 were found; however, the displacement 
was higher than in the rest of the cases. This result 
shows that the fibre orientation is not suitable to resist 
the working conditions of the RBP. The C3 presents 
the higher SF, but at the same time, the displacement 
was shorter with only 5 mm. The best balance 
between displacement and SF was found in C2. In 
this case, the displacement was 9.19, and the SF was 
1.34. Carbon fibres with 0°/90° orientation results in 
the best option for the design of RBP. The addition of 
layers with 0°/90° in a combined case, C3, reduces 
the capability of displacement of only layers oriented 
–45°/45. The fibres oriented at –45°/45 ° have a more 
elastic behaviour because they are not perpendicular 
to the direction of the applied load; therefore, there is 
energy dissipation.
Table 5.  Displacement and SF obtained from optimization
Displacement [mm] SF
C1 16.24 0.43
C2 9.19 1.34
C3 5 1.45
In Fig. 9, the displacement and SF obtained 
from the optimization were compared with those 
obtained in the database. The search space is reduced 
to the zone where one of the two requirements, 
either displacement or SF, stands a greater chance of 
improvement. In C1, the increase of displacement is 
remarkable due to the low capability of RBP design to 
increase its strength to the applied load. C2 presents 
the same trend to improve its displacement; however, 
its SF is over the defined limit, which favours the RBP 
design. In the case of C3, the trend is reversed and 
now the optimization leads toward and improvement 
of the SF, decreasing considerably its displacement. 
These results confirm C2 as the best option for the 
design of the RBP.
Fig. 9.  Comparison between database and optimized cases;  
C1, C2, and C3

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
97 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
The design variables for each case are presented 
in Table 6. It is observed that R2 presents the same 
value for the three cases and corresponds to its lower 
bound. It should be noted that W in C1 is on the low 
bound, and this had a negative impact in its resistance. 
Cases with a short number of layers were those who 
presented higher displacements, whereas in C3 with 
24 layers, the displacement was reduced. Then, it 
is observed that a lower number of layers helps to 
decrease the stiffness. The R2 segment, being in the 
middle part of the RBP in relation to its height and 
a greater horizontal distance from the direction of 
application of the load, is subjected to a greater 
bending moment than the rest of the segments of 
the RBP. For this reason, the smaller the dimension 
of R2, the better the behaviour of the prosthesis. The 
values of segments R1, R3, and R4 vary depending on 
maintaining an optimal horizontal extension between 
R2 and R4 to minimize the effect of total flexion of 
the RBP. Also, the fact that manufacturing costs are 
reduced with fewer layers must be considered.
Table 6.  Optimized design parameters
Design parameter C1 C2 C3
R1 [mm] 106.207 129.7 108.85
R2 [mm] 74.7 74.7 74.7
R3 [mm] 346.49 305.61 333.68
R4 [mm] 133.75 129.14 110.7
W [mm] 40.5 49.48 49.5
No. of layers 18 18 24
A comparison of the RBP profiles is presented 
in Fig. 10. At the segment R1, the profile of both C1 
and C3 is similar and changes from R2, whereas the 
C2 profile presents lower outward curvature. All the 
profiles match in the junction of the end of R3, and 
the start of the ground contact area due to the length 
of this area is constant. Beyond the contact area, the 
profiles separate again, and the C3 curves inward to 
RBP more than C2 and C3.
Fig. 10.  Comparison of optimized RBP profiles
The displacement contours of the optimized 
RBP are presented in Fig. 11. The RBP displacement 
is higher at the top and diminishes until the area of 
contact with the ground. Beyond this point until the 
bottom tip, the displacements are negligible. The 
difference of displacements between top and point of 
contact of the RBP optimized models indicates that the 
stiffness of C1 design is lower than C2 and C3, whereas 
C3 presents higher stiffness. Higher displacement is 
a characteristic that favours the return of energy that 
the RBP must offer when operating. The C1 design 
presents the highest displacement value due to the 
behaviour of the fibres oriented at –45°/45° that tend 
to dissipate energy. This level of displacement would 
be desirable if it is complemented by an acceptable 
SF. The C2 design with fibre orientations at 0°/90° 
offers a higher level of stiffness, which produces a 
lower displacement value than the C1 design; this is 
because part of the fibres of this configuration oppose 
resistance in a direction parallel to the applied force. 
The combination of –45°/45° and 0°/90° orientations 
further increases stiffness, affecting displacement, 
obtaining the lowest displacement values of the three 
designs.
Fig. 11.  Displacement contours of optimized cases;  
a) C1, b) C2, and c) C3
Fig. 12 shows the zones where the SF occurs 
in the RBP optimized designs and its magnitudes. 
The critical zone is the contact area in which the 
SF is lower. This occurs due to the fact that in this 
segment the displacements are restricted, and reaction 
forces increases. Zones with high curvature also 
present lower SF, which can be attributed to a stress 
concentration in these locations. In C1, due to the 
orientation of layers, low SF magnitude covers a 
large section of the RBP. Areas below the minimum 
(SF < 1) appear in the C1 design; therefore, under 
the analysis conditions, it is not satisfactory since 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
98 Rosel-Solís, M.J. – Dávalos, J.O. – Molina-Salazar, J.– Ruiz-Ochoa, J.A. – Gómez-Roa, A.
the RBP fails. The fibre orientation –45°/45° favours 
the flexural capacity of the RBP but acts negatively 
on its resistance. Designs C2 and C3 obtained 
satisfactory SF values, with C3 obtaining the best. 
However, considering the displacement results, the C2 
design is the one that offers a better balance between 
displacement and SF, in addition to requiring a lower 
number of layers than C3. The combination of fibre 
orientation in the C3 design increased its stiffness and 
decreased its flexibility, obtaining a higher SF but 
reducing displacement.
Fig. 12.  SF contours of optimized cases; a) C1, b) C2, and c) C3
Fig. 13.  Displacement and SF along RBP
Displacement and SF were obtained along a 
path at the centre of the RBP in Fig. 13. The length 
of the RBP was normalized from top to bottom. 
The maximum displacement is observed at the top 
of the RBP and remains constant. Near the middle 
displacement diminishes and drops to zero at the point 
of contact. Due to the amplitude of displacements, 
large forward propulsion will be expected in C1 but, 
due to fluctuations observed, the return of energy will 
be smoother in C2 and C3. In the case of SF, three 
critical zones were identified. In the design, C1 failure 
occurs in almost 25 % of the RBP length whereas in 
C2 and C3 the percentage of length with lower SF 
values is smaller. Another difference was the region 
where minimum SF appears in C1 is at 10 % away 
from the RBP top; cases C2 and C3, this region was 
near the point of contact.
3.4  Sensitivity Analysis Results
The results of the sensitivity analysis are presented in 
Fig. 14. The displacement is mostly influenced by the 
number of layers in all cases and to a lesser extent by 
R1 for the cases C1 and C3, whereas in the C2 case, 
the lowest percentage of relative importance was 
the R4. In the SF, the number of layers is the most 
important variable for C1 and C3; however, in C2, it 
is the width of the prosthesis. These differences can be 
attributed to the orientation of the fibres. The fibres at 
–45°/45° make the RBP more flexible but decreases 
its resistance; therefore, to improve this negative 
condition, the carbon fibre layers must be increased. 
In the case of the 0°/90° fibre orientation design, by 
increasing the width of the RBP, the number of fibres 
acting to support the applied load is increased, which 
improves their resistance and displacement.
  
 
 
Fig. 14.  Relative importance of the design parameters;  
a) C1 displacement, b) C1 SF, c) C2 displacement,  
d) C2 SF, e) C3 displacement, f) C3 SF

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)3, 88-100
99 Optimization of Running Blade Prosthetics Utilizing Crow Search Algorithm Assisted by Artificial Neural Networks 
4  CONCLUSIONS
An optimization procedure that includes CSA and 
ANN was employed to optimize the design of an RBP 
with the objective of increasing its displacement to 
prevent failure through the Tsai-Wu failure criterion. 
The RBP was made of carbon fibre layers stacked 
around a core made of ABS. Three carbon fibre layer 
orientations were tested. For the design of the RBP , the 
results show that the layers oriented at 0°/90° represent 
the best trade-off between displacement and Tsai-Wu 
criterion with a reduced number of layers, whereas 
–45°/45° are not suitable due to imminent failure. The 
case with combined orientations presented minimal 
displacement, which would increase its stiffness. The 
damage was concentrated in the point of contact with 
the ground and, due to the selection of adequate design 
parameters, was minimized in the zones of the RBP 
with a curved shape. The two most important design 
parameters were the number of layers and the width of 
the RBP. The developed optimization procedure that 
combines CSA and ANN reveals a fast technique to 
improve the design of RBP; in addition, it is feasible 
to use the proposed manufacture technique for future 
orthopaedic personalized applications.
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