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Citation: Cortés Torres, E.d.J.; García
Gonzales, L.E.; Villamizar Marin, L.E.;
García Cena, C.E. Mechanical Design
of a New Hybrid 3R-DoF Bioinspired
Robotic Fin Based on Kinematics
Modeling and Analysis. Actuators
2024,13, 353. https://doi.org/
10.3390/act13090353
Academic Editor: Zhuming Bi
Received: 10 August 2024
Revised: 8 September 2024
Accepted: 9 September 2024
Published: 11 September 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
actuators
Article
Mechanical Design of a New Hybrid 3R-DoF Bioinspired
Robotic Fin Based on Kinematics Modeling and Analysis
Eliseo de J. Cortés Torres *,† , Luis E. García Gonzales, Luis E. Villamizar Marin and Cecilia E. García Cena
Escuela Técnica Superior de Ingeniería y Diseño Industrial, Centre for Automation and Robotics (UPM-CSIC),
Universidad Politécnica de Madrid, Ronda de Valencia, 3, 28012 Madrid, Spain;
le.garcia@alumnos.upm.es (L.E.G.G.); luis.villamizar@alumnos.upm.es (L.E.V.M.);
cecilia.garcia@upm.es (C.E.G.C.)
*Correspondence: eliseo.cortes@alumnos.upm.es
†Current address: Ronda de Valencia, 3, 28012 Madrid, Spain.
Abstract: The field of bioinspired underwater robots aims to replicate the capabilities of marine
animals in artificial systems. Stingrays have emerged as highly promising species to be mimicked
because of their flat body morphology and size. Furthermore, they are considered high-performance
species due to their maneuverability, propulsion mode, and sliding efficiency. Designing and devel-
oping mechanisms to imitate their pectoral fins is a challenge for underwater robotic researchers
mainly because the locomotion characteristics depend on the coordinated movement of the fins. In
the state of the art, several mechanisms were proposed with 2 active rotation degrees of freedom
(DoFs) to replicate fin movement. In this paper, we propose adding an additional active DoF in order
to improve the realism in the robotic manta ray movement. Therefore, in this article, we present
the mechanical design, modeling, and kinematics analysis of a 3-active-and-rotational-DoF pectoral
fin inspired by the Mobula Alfredi or reef manta ray. Additionally, by using the kinematics model,
we were able to simulate and compare the behaviour of both mechanisms, that is, those with 2 and
3 DoFs. Our simulation results reveal an improvement in the locomotion, and we hypothesized that
with the third DoF, some specific missions, such as hovering or fast emergence to the surface, will
have a better performance.
Keywords: bioinspired; compliant joints; underwater robots
1. Introduction
For millions of years, different species have evolved in such a way that they have
adapted to the diverse conditions of their environment, with an example being fish and
marine animals, which have developed behavioral, physiological, and morphological
adaptations to face the conditions of the marine environment, specifically the temperature,
salinity, pH, oxygen and carbon dioxide contents of the water, and pressure, among others,
which, in many cases, can become extreme [
1
]. Numerous researchers have been interested
in studying these adaptations in order to understand natural behaviour and describe
the hydrodynamic principles that drive them. This knowledge has been used in various
applications, including bioinspired underwater robotics.
Biologically inspired underwater vehicles are designed by imitating the characteristics
of marine species, such as their high-efficiency capabilities [
2
]. Batoid fish, including
stingrays, are considered high-performance species due to their maneuverability and thrust
production, allowing them to leap out of the water and reach speeds close to 2.8 m/s
with low frequencies of flapping and undulating. Thanks to these characteristics, they can
migrate over long distances, populate underwater regions, and even descend to depths of
around 1400 m [
3
,
4
]. Furthermore, their flat body and large size can be replicated to enhance
the payload capability, and their pectoral fins can offer vectored propulsion, providing
Actuators 2024,13, 353. https://doi.org/10.3390/act13090353 https://www.mdpi.com/journal/actuators
Actuators 2024,13, 353 2 of 13
great sliding efficiency typical of median and paired pectoral fin (MPF) mode, allowing
them to travel long distances without resting [5–7].
Given the aforementioned characteristics, numerous research groups have devel-
oped underwater vehicles inspired by stingrays [
6
,
8
,
9
]. Furthermore, several studies have
focused on important aspects such as the hydrodynamics of stingrays [
10
] to better un-
derstand the locomotion of these species, as well as the use of different materials for their
construction [
11
–
14
], including the use of actuators with smart materials, such as shape
memory alloy wire (SMA) [
15
,
16
]. However, the main emphasis lies in the development
of suitable mechanisms to improve realism in robotic fin locomotion. The MFP swim is
characterized by maneuverability and stability. For vehicles inspired by stingrays, it is
essential to use mechanisms capable of executing oscillating movements of the pectoral
fins and achieving adequate propulsion. Mechanisms have been developed based on
propulsion speed, thrust, and robot power, presenting different configurations with con-
trollable parameters, namely frequency, amplitude, wavelength, fin shape, and undulatory
mode [
17
]. Some of the developed configurations include a radial scissor mechanism with
only one actuator per fin [
18
], a propulsion mechanism with undulating fins with multiple
actuators [
19
], a rocking crank mechanism [
20
], and flexible beam mechanisms achievable
by implementing metal cables and rigid bones [21], among others.
In the case of stingrays, the muscles of the fins are thicker, while those at the basal part
are stronger, generating greater driving power. In the distal part, there is more cartilage
and soft tissue, which serve as controllers to adjust the wave transfer on the fin surface.
In [
5
], a pectoral fin module is proposed with a 2 DoF spatial parallel mechanism (SPM)
that mimics these characteristics.
Morphology Selection
Among the Mobula species, there exist different specimens that could be used as
inspiration for designing underwater robots, including devil manta rays, reef manta rays,
and giant manta rays. According to [
22
], most of them live in the world’s extensive water
areas, which helps with integrating the robot into the environment. Manta rays’ glider
shape offers different advantages; for instance, some Mobula alfredi specimens can reach a
wingspan of 8.8 m [
23
]. This characteristic is considered as a strength in the selection of
this type of animal, and the size of its body is suitable for the integration of mechatronics
elements. Indeed, small objects in the ocean are difficult to identify, but the manta ray shape
is suited to stealth tasks. Furthermore, the natural flexibility of the manta ray fins and its
glider shape also help to facilitate greater manoeuvrability in small spaces. Additionally,
these marine animals can navigate over 450 m of depth, another characteristic making their
morphology suitable for underwater robot applications.
Based on previous considerations, we considered the morphology of a manta ray
(Mobula Alfredi) as the baseline for the mechanical design of our bioinspired underwa-
ter robot.
Other authors have also considered a manta ray morphology. Perhaps the mechanism
proposed by [
5
] is the closest to our mechanism. However, in [
5
], the locomotion pattern
is generated by the SPM placed near to the fin base. These authors suggested that an
additional DoF could improve the behaviour of the fin, making it more natural. In [
21
,
24
],
the authors proposed different mechanisms for the fin, wherein the SPM was replaced by a
fin designed under cantilever beam principles.
In this paper, we analyze the impact of the third DoF in terms of movement ampli-
tude and power consumption by modeling an entire mechanism and simulating it under
different scenarios.
According to the evidence provided by our theoretical study, we decided to include an
additional DoF in the mechanical design. Additionally, in this article, we present a proto-
type of a new hybrid pectoral robotic fin based on a Mobula Aldredi manta ray with 3 active
rotational DoFs. Despite, the third degree of freedom introducing additional complexity in
kinematic modeling, the hypothesis is that it could reduce the power consumption while
Actuators 2024,13, 353 3 of 13
improving the performance and efficiency of the navigation by making the movement more
natural. In addition, the manoeuvrability could be improved due to this third DoF, mainly
targeting manoeuvrability in narrow spaces. We have also compared the strengths and
weaknesses of both morphologies, that is, those with 2 and 3 DoFs, in order to validate the
hypothesis that an additional DoF improves the navigation capabilities of a bioinspired
fin, and in some missions, hovering could be optimized by using this extra DoF. Figure 1a
shows a model sketch using the manta ray as a background for design.
(a) (b)
Figure 1. The 3 DoF robotic fin mechanism and the scheme for the kinematics model: (a) hybrid
mechanism coupled with a background representation of a manta ray (Mobula Alfredi) and (b) spatial
hybrid mechanism representation with the 3 DoFs, where joints I, V, and VI are active, while joints II,
III, and IV are passive.
This document is organized as follows: Sections 2and 3present the analysis and
simulation of the kinematic model of the mechanism with 2 and 3 DoFs, and finally,
Section 4presents the mechanical design.
2. Kinematic Analysis
Figure 1b presents the configuration of the spatial hybrid mechanism with 3 active
degrees of freedom. Furthermore, the figure shows the local reference systems associated
with each joint.
Using a Lie algebra methodology, the entire mechanism is split up; then, it can be ana-
lyzed by considering two subsystems, with one of them involving the spatial mechanism,
S0
to
S6
, with 2 DoFs, and the other involving the 1 DoF serial mechanism described by
S6
to STf. In the following subsections, each subsystem is mathematically modeled.
2.1. Two-Degrees of Freedom Spatial Mechanism Direct Kinematics
The method used to analyze the joint localization of the mechanism is based on the Lie
algebra method and is extensively explained in [
25
]. Therefore, an
i
th coordinate system is
created on each
i
th body, aiming to describe its position with respect to a global system
{S0}
. This approach suggests the transformation of close-loop mechanisms into open-loop
mechanisms restricted for the position and orientation in a common joint where all the
kinematic chains converge [
25
]. Thus, it is possible to define two vectors, one for the active
joints (Equation (1)), and the other for the passive joints (Equation (2))
θact = [θ5,θ1], (1)
θpas = [dz4,θ2,dz3]. (2)
where
θi
and
dzi
are the variables associated with the rotational and prismatic joints, re-
spectively. In order to transform the close-loop mechanisms into open loop mechanisms,
the proposed method is applied to the spherical joint at the intersection of the rotational
coordinate systems
{Sm}
and
{Sn}
(see Figure 1b). Therefore, the spherical joint is decou-
pled in two kinematic chains, the first one between the systems
{x0
,
y0
,
z0}
to
{xm
,
ym
,
zm}
,
Actuators 2024,13, 353 4 of 13
and the second between the systems
{x0
,
y0
,
z0}
to
{xn
,
yn
,
zn}
. As a result, a homogeneous
transformation matrix (HTM) is made between {x0,y0,z0}and {x5,y5,z5}:
T0,5 =I rCM
0 1 Rz(θ5)0
0 1 (3)
where
rCM ∈R3×1
is the position vector between a global position (CM) and the
S1
and
S5
.
For the systems {x5,y5,z5}and {x4,y4,z4}, the HTM will be as follows:
T5,4 =Ry(π/2)0
0 1 I rz(l5+dz4)
0 1 (4)
where
rz
is the position vector cylindrical joint in
{S4,5}
. For the systems,
{x4
,
y4
,
z4}
and
{xm,ym,zm}
T4,m=I rz(l4)
0 1 (5)
On the other hand, the second kinematic chain relates the reference systems between
{x0
,
y0
,
z0}
and
{xn
,
yn
,
zn}
. Then, for the first system between
{x0
,
y0
,
z0}
and
{x1
,
y1
,
z1}
,
the homogeneous transformation matrix is given by:
T0,1 =I rCM
0 1 Rz(θ1)0
0 1 (6)
Between the reference systems
{x1
,
y1
,
z1}
and
{x2
,
y2
,
z2}
and for
{x2
,
y2
,
z2}
and
{x3
,
y3
,
z3}
, the transformations are defined by Equation (7) and Equation (8), respectively;
T1,2 =I rx(l1)
0 1 Ry(π/2)0
0 1 Rz(θ2)0
0 1 (7)
where rxis the position vector between joints 1 and 2 in {S1,2}. For the system between,
T2,3 =Ry(π/2)0
0 1 I rz(l2+dz3)
0 1 (8)
Finally, between {x3,y3,z3}and {xn,yn,zn},
T3,n=I rx(l3)
0 1 (9)
2.1.1. Constraint Vector
The behavior of the mechanism is described by the position analysis of the spherical
joint and by the physical constraints in the joint that has been decoupled. Spherical joints
have 3 DoFs related to the rotation in each axis. The constraints are related to the linear
displacement in the three axes and are described in the following equation:
Φ=
R0,nux,x(rn−rm)
R0,nuy,y(rn−rm)
R0,nuz,z(rn−rm)
(10)
where
ux,x
,
uy,y
,
uz,z
are unit vectors associated with the local axis in each joint, while
(rn−rm)
is the difference in position between the systems
{xn
,
yn
,
zn}
and
{xm
,
ym
,
zm}
,
since we have a base system view.
Actuators 2024,13, 353 5 of 13
2.1.2. Jacobian Matrix
The constraint vector time derivative is given by
˙
Φ=Φθ˙
θj
, which can be expressed
in Lie algebra terms as:
Φθ(j,i) = CT
nj,n j AdTn j,iSi(11)
Here, Siis an element of the Lie algebra se(3), which describes the velocity of the ith joint.
The covector
CT
nj,n j
represents the reaction force (wrench) at the decoupled joint, while
AdTnj,i
is the adjoint matrix, which represents the action of the Lie group
SE(
3
)
on its Lie
algebra
se(
3
)
. In general terms, the components
CTnj,n jAdTn j,iSi
describe the relationship
between the
j
th constraint and the
i
th joint’s DoF, and
˙
θi
represents the joint velocity values.
Thus, the twist described for the Jacobian
(Φθ)
of each passive and active joint can be
expressed by:
Srot =[0, 0, 1, 0, 0, 0]
Strans =[0, 0, 0, 0, 0, 1](12)
where
{Srot }
represents the twist description for the rotational joints, that, is axes
{S2}
,
{S4}
and
{S5}
, while
{Strans}
represents the twist for the prismatic joints, that is,
{S3}
and
{S6}. Finally, the Jacobian is given by,
Φθ(:, 1 : 2) = "−CT
xAdTn ,mAdTm ,5S5−CT
xAdTn ,mAdTm ,4S4
−CT
yAdTn ,mAdTm ,5S5−CT
yAdTn ,mAdTm ,4S4
−CT
zAdTn ,mAdTm ,5S5−CT
zAdTn ,mAdTm ,4S4#
Φθ(:, 3 : 5) = "CT
xAdTn ,1S1CT
xAdTn ,2S2CT
xAdTn ,3S3
CT
yAdTn ,1S1CT
yAdTn ,2S2CT
yAdTn ,3S3
CT
zAdTn ,1S1CT
zAdTn ,2S2CT
zAdTn ,3S3#
The Jacobian can be split, taking into account the passive and active joints, as follows:
Φθact =[Φθ(:, 1),Φθ(:, 3)](13)
Φθpas =[Φθ(:, 2),Φθ(:, 5),Φθ(:, 6)](14)
2.1.3. Position Analysis
The following (Equation (15)) describes the kinematic relation between active and
passive joints:
Φθact∆θact +Φθpas∆θpas =−Φ(15)
In the case that the parameters of the active joints are the reference signals for the
system, it could be assumed that ∆θact =0. Then, Equation (15) can be rewritten as
Φθpas ∆θpas =−Φ(16)
The Newton–Raphson method was implemented to obtain a numerical solution for
Equation (16).
2.2. One-Degree of Freedom Serial Mechanism Direct Kinematics
Due to this mechanism’s serial nature, the fin tip (
Tf
) position can be described by a
linear combination of its joints, as shown in Figure 2.
Actuators 2024,13, 353 6 of 13
Figure 2. Fin tip location change when a force
F
is applied, generating a deformation angle
Ψ
that is
considered as the third DoF.
Thus, for a given coordinate system
{XYZ}
located at
{S6}
and if all joints are
considered as rotational (see Figure 2), the position of
Tf
could be computed by the
following
To−x=
n
∑
i=1
HicosiΨ
n
To−y=
n
∑
i=1
HisiniΨ
n
. (17)
where
n=
3 is the number of passive DoFs of the mechanism,
H∈R3×1
is the vector
component for each link, and
Ψ
is the deformation or deflection angle created for the force
in the wires. This vector could be described as
H= [H1...Hn](18)
2.3. Inverse Kinematics
For the inverse kinematic analysis, the
α
and
β
angles are the input values required to
compute the inverse kinematics. Moreover, the
α
angle can be expressed as a function of
θ5
and θ1(see Figure 3).
For the spatial mechanism, the parameters
α
and
β
can be modeled using the following
equations:
α(t) = αmax Sin(2πf t)(19)
β(t) = βmaxCos(2πf t)(20)
where
f
is the frequency of the movement,
t
is the time domain variable, and
βmax
and
αmax
are related to the mechanism-desired motion, considering their physical constraints. There-
fore,
βmax
is the maximum pitching angle, and
αmax
is the maximum flapping amplitude.
Considering Figure 3, it is possible to express the αangles as:
α(t) = (θ1+θ5)/2, (21)
and
arctan(l3/l1) = θ5−θ1(22)
where
l3=l0tan(β(t)) (23)
Actuators 2024,13, 353 7 of 13
Considering Equations (21) and (22), variables θ5and θ1could be computed by
θ5= (arctan(l3/l1) + 2α)/2 (24)
θ1= (2α)−θ5(25)
Finally, for the serial mechanism,
n
∑
i=1
iΨ
n=α(26)
Figure 3. The geometricrelationship parameters considering the mechanism motion plane.
3. Kinematic Analysis Simulation
In this section, we describe how we conducted simulations to analyze whether the
additional DoF improves the mobility of the fin.
To facilitate this, we implemented a previous kinematic model into a MatLab environ-
ment. Four different scenarios are proposed. In each scenario, the input signal is modified
while the extra DoF is switched on and off in order to evaluate the improvements in the
movements in terms of amplitude.
For Scenarios 1 and 2, the CPG input signal, according to [
5
], was implemented.
In Scenario 1, the third DoF is switched off, while in Scenario 2, it is switched on. We
proceed in a similar way for Scenarios 3 and 4, but in these cases, the input signal is
mathematically described by [
10
] and replicates the natural movements of a manta ray. In
Scenario 3, the third DoF is switched off, while in Scenario 4, it is switched on.
Tables 1and 2show the physical parameters and mechanical constraints for links and
joints according to the mechanical design.
Table 1. Geometric considerations implemented in simulation.
ith Link Length (mm)
L1104
L5104
L2172
Actuators 2024,13, 353 8 of 13
The simulation time is
t=
6.6 s , and the frequency is
f=
0.15 Hz. Additionally,
the maximum pitching angle is
βmax =
25
◦
, and the maximum flapping amplitude is
αmax =60◦.
Table 2. Translational joints’ constraint limits.
ith Joint Max. Displacement (mm)
dz438
dz340
3.1. Simulations Using a Sinusoidal CPG Approach
Figure 4a shows the fin mechanism movement, where the point
Tf
describes a curve
similar to a linear arc, while the actuated joints reach values around
θ1,5 >
50
◦
and
θ1,5 <−
50
◦
respectively. For the second scenario, Figure 4b, the input parameters are
on hold, while the additional DoF is switched on. Then, due to the addition of a serial
mechanism, the target flapping amplitude αincludes the α1and α2angles.
The angle
α1
is implemented by Equation (21), which is related to the spatial mech-
anism, while the
α2
angle is implemented by Equation (26), which is associated with the
serial mechanism. Thus, the target flapping amplitude αis described by:
α=α1+α2(27)
Here,
α1=
0.4
α
, and
α2=
0.6
α
. Additionally, for this simulation,
n=
3, the vector
H= [
34, 25, 35
+
296
]
, and
Ψ=α2
. From Figure 4b, the change in the shape of the curve
described by the fin’s tip can be observed. It looks more similar to the natural manta ray
movement, that is, the ∞shape.
By comparing both scenarios, we found that the values computed for the actuated
joints are lower when the third DoF is switched on: θ1,5 >25◦and θ1,5 <−25◦.
(a) (b)
Figure 4. Sinusoidal CPG input according to [
5
]. (a) Scenario 1: a literature approach with a 2 DoF fin
mechanism. (b) Scenario 2: a mechanism with 3 DoFs, proposed by the authors.
3.2. Simulations Using a Natural Manta Ray Movement Approximation Approach
An approximation of a natural manta ray movement is defined by [
10
] as a relationship
of two main variables defining the articulations trajectories as follows:
α(t) = 206.7 ×sin0.037 ×60 ×(t+ϕ)
π+2.40+
176.3 ×sin0.033 ×60 ×(t+ϕ)
π−0.5+
10.07 ×sin0.098 ×60 ×(t+ϕ)
π−0.29
(28)
Actuators 2024,13, 353 9 of 13
β(t) = 34.67 ×sin0.048 ×60 ×(t+ϕ)
π+1.073+
9.08 ×sin0.107 ×60 ×(t+ϕ)
π−1.783+
7.14 ×sin0.148 ×60 ×(t+ϕ)
π−2.41
(29)
Equations (28) and (29) are considered for the fin mechanism simulation, taking into
account the maximum pitching angle
βmax ≤
30
◦
and the maximum flapping amplitude
αmax ≤
60
◦
. However, due to the size discrepancy between a real manta ray and the
prototype proposed, it is not possible to implement these input variables without modifying
their amplitudes. Thus, the new pitching angle is
βn=
0.7
β(t)
, and the new flapping
amplitude is αn=0.72α(t)for the system without the third DoF.
In Figure 5, the simulation results derived from when the input signal can be given by
Equations (28) and (29)
are presented. Indeed, Figure 5a shows the performance of the fin
mechanism when the third DoF is switched off. The curve described by the end-effector
Tf
is, again, shaped like a linear arc due to the fin stiffness. The parameters related to the
actuated joints are similar to those obtained with the CPG approach: −50◦>θ1,5 >50◦.
For Scenario 4, the parameters selected for the simulation allow us to split the flap-
ping amplitude between both mechanisms:
α1=
0.3
α(t)
and
α2=
0.7
α(t)
. Due to the
mechanism combination, it is not necessary create a new value for αn.
Figure 5b shows the results for Scenario 4, that is, a scenario wherein the third DoF is
switched on. The movement of the end-effector,
Tf
, describes an elliptic curve, mainly due
to the system capacity, to split the flapping amplitude. Moreover, the parameters associated
with active joints are lower than in Scenario 3, that is, θ1,5 >25◦and θ1,5 <−25◦.
(a) (b)
Figure 5. Input signal according to [
10
]; manta ray natural movement; (a) Scenario 3: simulated
movement with the third DoF switched off; (b) Scenario 4: the authors’ proposed mechanism with
the third DoF switched on.
4. Three-Dimensional Prototype of the Robotic Fin
Based on the kinematics analysis, the design of the robot is proposed as a combination
of two systems to test the concept. The first system is a 2 DoF spatial mechanism, and the
second system is a 1 DoF mechanism based on a cantilever beam with a middle compliant
joint using layers of plastic materials; the hybrid mechanism decoupled is shown in Figure 6.
The main construction material used was acrylonitrile styrene acrylate (ASA), and
construction was carried out using Fused Deposition Modeling (FDM) techniques for
fabrication, while metallic parts were built using lathe techniques.
4.1. Prototype of 2 DoF Spatial Mechanism
Based on the design proposed by [
5
], a first prototype was designed to validate the
movement. The design of the rear link was based on a commercial ball joint from SKF; this
joint allows for a movement of 14
◦
in all their axes. For the design of the lateral link, we
Actuators 2024,13, 353 10 of 13
used a NACA0022 profile. However, this profile was modified to enhance the range of
movement of the spherical joint. Additionally, due to the redundancy between cylindrical
and spherical joints, it could be used to improve the workspace of the spherical joint.
Moreover, due to the size of the spherical joint body, a mechanical coupling was
designed and manufactured to adapt the minimum distance between the lateral link base
and the rear link (see Figure 6a).
(a) (b)
Figure 6. Computer-Aided Design (CAD) description of the hybrid mechanism: (a) 3D model design
proposed for the 2 DoF SPM; (b) proposed 3D model that is inclusive of the third DoF and based on a
cantilever beam using compliant joints and wire to move it.
For this prototype, linear bearings, instead of friction bearings, were used due to
blocking in the movements, lack of alignment, and the friction problems between the rear
link and the rear shaft. In the lateral link, a compliant joint is placed at the 1 mm slot.
It remains attached to the main body, and another slot is designed in order to insert a
servomotor that transmits the power to the third degree of freedom (see Figure 6a).
4.2. Prototype of 1 DoF Mechanism
Despite the fact that, regarding the state of the art, it is possible to find previous re-
search articles aimed at replicating manta ray movement, like [
10
,
24
,
26
], the main difference
between our proposal and those previously reported lies in the actuation system of the
additional DoF. Our mechanism was designed to be driven by wires. Then, the properties
of plastic materials are used to emulate a compliant joint with four ribs, being part of the
fin. The base, i.e., the lateral link base, with its curved-shape design, supports the forces
applied to the wires.
For the real mechanism validation, a 1 mm polyethylene terephthalate glycol (PETG)
sheet was selected; however, for real application, a thickener sheet will be used to prevent
natural flexion due to the weight of the fin tip and the gravity forces applied on it. The shape
selected covers the biggest area to support the weight of the entire mechanism. Despite the
mechanism designed by [
26
] being 10 mm, our design is bigger due to the space between
the base and the first rib.
Finally, the last part of the cantilever beam is made of a rigid carbon fiber the pattern
of which replicates the fin shape. It is well known that carbon fiber material has good
mechanical properties, such as stiffness and flexibility. The fin will be covered by a suitable
skin made of polychloroprene or silicone.
The result of merging these two mechanisms is shown in Figure 7a.
Actuators 2024,13, 353 11 of 13
(a) (b)
Figure 7. Pictures of the developed fin prototype: (a) real prototype based on a hybrid robot.
(b) Movement generated when applying the force described in the real prototype.
5. Discussion
In this paper, we have presented a new hybrid 3R-DoF mechanism aimed at replicating
the natural movement of a manta ray fin. Despite similar approaches existing in the
literature, our design introduces a novel aspect in the actuation system of the third DoF that
is made by a wire that passes through ribs and pulls them up and down, and as consequence,
the movement becomes soft because of this compliant joint. Additionally, the kinematics
model was proposed under the Lie algebra method, which allows one to decouple the
3 DoF mechanism into two mechanisms, one with 2 DoFs and one with 1 DoF. After the
implementation of the model in Matlab environments, several simulations were performed
in order to compare the behaviour of the mechanisms proposed in the literature with 2 DoFs
with our mechanism.
The natural undulatory movement was reproduced by a sinusoidal CPG. According
to our results, the proposed mechanism improves the locomotion compared to those
reported previously in the literature. Therefore, the 3R-DoF was designed and prototyped
considering the hybrid mechanism SPM+compliant joint. Furthermore, the angle needed to
obtain a full undulatory trajectory is lower. Taking into account the simulation results, it is
possible to increase the autonomy of the robot due to the reduction in energy consumption
by the actuators of the spatial parallel mechanism . On the other hand, manta rays have the
ability to move in narrow spaces; this means that with the addition of the third DoF, it is
possible to achieve high-performance maneuvers with small movements.
Kinematic analysis offers a deep understanding of how the addition of an extra DoF
affects the whole mechanism. According to this, the transformation of
αmax
into
α1
and
α2
supposes a reduction in the angular displacements in the actuated joints, allowing for
optimal values of movements, increasing the workspace.
Finally, we also conducted a study on whether the spherical joint mobility constraint
affects the system movements and whether re-designing could increase the maximum
pitching angle βma x.
6. Conclusions
In this paper, the design, prototyping, kinematics modeling, and simulation of a 3R-
DoF hybrid mechanism was presented. The fin part of the manta ray underwater robot
is currently under development. In this paper, we presented theoretical and simulation
results that allow us to compare the performance of 2 DoF fin mechanisms, widely reported
in the literature, and our mechanism, which was made with 3R-DoF and a compliant
joint. The evidence presented demonstrates that the additional degree of freedom can
contribute to improving the maneuverability and, at the same time, increase the autonomy
of the underwater robot. Because the prototype was developed to turn the third DoF off or
on, the next step is to carry out experiments in water to validate the advantages that the
additional DoF provides to the robot.
Actuators 2024,13, 353 12 of 13
Author Contributions: Conceptualization and writing—original draft, E.d.J.C.T. and C.E.G.C.;
conceptual design, E.d.J.C.T. and C.E.G.C.; formal analysis, E.d.J.C.T. and L.E.G.G.; methodol-
ogy, E.d.J.C.T. and C.E.G.C.; coding, E.d.J.C.T. and L.E.G.G.; simulation, E.d.J.C.T. and L.E.G.G.;
results, E.d.J.C.T. and L.E.G.G.; figure creation, E.d.J.C.T.; validation, E.d.J.C.T. and L.E.G.G.; writing—
reviewing and editing, E.d.J.C.T., L.E.G.G., L.E.V.M. and C.E.G.C.; funding acquisition, C.E.G.C. All
authors have read and agreed to the published version of the manuscript.
Funding: This research was partially supported via a research project, Chair University-Industry
Monodon, by Navantia Underwater Robots and Deeptech Technology. Reference id: CAT235618000.
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
DoF Degree of freedom
MPF median and paired pectoral fin
SMA shape memory alloy wire
SPM spatial parallel mechanism
HTM homogeneous transformation matrix
CM global position
CPG Central Pattern Generator
ASA acrylonitrile styrene acrylate
FDM Fused Deposition Modeling
CAD Computer-Aided Design
PETG polyethylene terephthalate glycol
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