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Article
Structural-Parametric Synthesis of the RoboMech Class Parallel
Mechanism with Two Sliders
Zhumadil Baigunchekov 1,2, Med Amine Laribi 3, * , Giuseppe Carbone 4, Azamat Mustafa 2, *, Bekzat Amanov 1
and Yernar Zholdassov 1
Citation: Baigunchekov, Z.; Laribi,
M.A.; Carbone, G.; Mustafa, A.;
Amanov, B.; Zholdassov, Y.
Structural-Parametric Synthesis of the
RoboMech Class Parallel Mechanism
with Two Sliders. Appl. Sci. 2021,11,
9831. https://doi.org/10.3390/
app11219831
Academic Editor: Manuel Armada
Received: 15 September 2021
Accepted: 14 October 2021
Published: 21 October 2021
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4.0/).
1Department of Mechanics, Al–Farabi Kazakh National University, Almaty 050040, Kazakhstan;
bzh47@mail.ru (Z.B.); bekzat.amanov007@gmail.com (B.A.); jera.kz@mail.ru (Y.Z.)
2Department of Mechanical Engineering and Modeling, Satbayev University, 22a Satpaev Str.,
Almaty 050013, Kazakhstan
3
Department GMSC, Institut PPRIME, Universitéde Poitiers, CNRS, ENSMA, UPR 3346, 86962 Poitiers, France
4
Department of Mechanical, Energy and Management Engineering, University of Calabria, 87036 Rende, Italy;
giuseppe.carbone@unical.it
*Correspondence: med.amine.laribi@univ-poitiers.fr (M.A.L.); mustafa_azamat@mail.ru (A.M.)
Abstract:
This paper addresses the structural-parametric synthesis and kinematic analysis of the
RoboMech class of parallel mechanisms (PM) having two sliders. The proposed methods allow
the synthesis of a PM with its structure and geometric parameters of the links to obtain the given
laws of motions of the input and output links (sliders). The paper outlines a possible application
of the proposed approach to design a PM for a cold stamping technological line. The proposed
PM is formed by connecting two sliders (input and output objects) using one passive and one
negative closing kinematic chain (CKC). The passive CKC does not impose a geometric constraint
on the movements of the sliders and the geometric parameters of its links are varied to satisfy the
geometric constraint of the negative CKC. The negative CKC imposes one geometric constraint on the
movements of the sliders and its geometric parameters are determined on the basis of the Chebyshev
and least-square approximations. Problems of positions and analogues of velocities and accelerations
of the considered PM are solved to demonstrate the feasibility and effectiveness of the proposed
formulations and case of study.
Keywords:
parallel mechanism; RoboMech; structural-parametric synthesis; Chebyshev and least-square
approximations; kinematic analysis
1. Introduction
The design of manipulation robots both with serial and parallel manipulators is
carried out mainly by solving inverse kinematics and developing the control systems and
technical means according to the obtained laws of motions of the actuators [
1
–
4
]. In this
case, the actuators of manipulation robots may work in controlled regimes of intensive
accelerations and braking that worsen their dynamics and mechanical efficiency [5,6].
In order to improve the dynamic characteristics and simplify the control systems
of the designed manipulators, it is advisable to set the laws of motions of the actuators
along with the given laws of the end-effectors’ motions. The ability to set the laws of
motions of the input links improves the dynamic parameters and simplifies the control
system and therefore also increases the reliability and reduces the cost of the designed
manipulator. Such parallel manipulators, having the property of manipulation robots, such
as reproducing the given laws of motions of end-effectors, and the property of mechanisms,
such as setting the laws of motions of actuators, are called the RoboMech class parallel
mechanisms or paralell manipulators (PMs) [7].
In the simultaneous setting the laws of motions of the input and output links, the RoboMech
class PMs operate under certain structural schemes and geometric parameters of their links.
Appl. Sci. 2021,11, 9831. https://doi.org/10.3390/app11219831 https://www.mdpi.com/journal/applsci
Appl. Sci. 2021,11, 9831 2 of 17
In this case, the control elements for the movement of the PMs, i.e., the functional relation-
ship between the laws of motions of the input and output links, is laid in the determining
structure schemes and geometric parameters of the links, i.e., in the mechanical part of the
RoboMech class PMs. Such an optimal combination of mechanics and motion control of
manipulation robots corresponds to the modern concept of mechatronics as a methodology
for developing of simple, reliable and cheap technological automation.
The base for the structural synthesis of planar mechanisms is proposed by Assur [
8
],
according to whom the mechanism is formed by connecting to the input link (actuator)
and the base of structural groups with zero degrees of freedom (DOF). These structural
groups are then called the Assur groups, which can be of different classes and orders.
The existing methods of structural synthesis of mechanisms and manipulators are devoted
to the determination of their structural schemes according to the given numbers of DOFs,
links, kinematic pairs and their types [
9
–
15
]. A review of research on the synthesis of types
of parallel robotic mechanisms was undertaken in [
16
]. All of these methods do not take
into account the functional purposes of mechanisms or manipulators.
In kinematic synthesis (dimensional or parametric synthesis) of mechanisms, with their
known structural schemes, the synthesis parameters are determined by the given positions
of the input and output links. Generation of the specified movements of output links
(output objects) can be performed exactly and approximately. Exact reproduction of the
required movements of a rigid body by linkage mechanisms is possible with a limited
number of positions, depending on the structural scheme of the mechanism-generator,
while the possibility of their approximate reproduction is not limited to the number of
specified positions.
Exact methods for synthesis of mechanisms, or so-called geometric methods, are based
on kinematic geometry. The fundamentals of kinematic geometry for finite positions of
a rigid body in a plane motion were developed by Burmester and for finite-positions of
a rigid body in space were developed by Shoenflies. Burmester in [
17
] developed the
theory of a moving plane having four and five positions on circles. Shoenflies in [
18
]
formulated theorems on the geometrical places of points of a rigid body having seven
positions on a circle and three positions on a line. The graphical methods of Burmester and
Schoenflies theories received an analytical interpretation [
19
–
21
], which is summarized in
the monograph.
Geometric methods of mechanism synthesis are clear and simple. However, these
methods are applicable only for a limited number of positions. Moreover, the algorithms
for solving problems using these methods depend significantly on the number of specified
positions, and their complexity increases with the number of positions. Approximation
(algebraic) methods of mechanism synthesis are devoid of these disadvantages.
Problems of approximation synthesis of mechanisms were first formulated and solved
in [
22
]. Least-square approximations are the most widely used in the approximation
synthesis of mechanisms. For the development of this method, a new deviation func-
tion, a weighted difference with a parametric weight, proposed in [
23
], was important.
In contrast to the actual deviation, the weighted difference can be reduced to linear forms
(generalized polynomials). This makes it quite easy to apply linear approximation methods
to the synthesis of mechanisms. This eliminates the limit on the maximum number of
specified positions of the moving object.
Combining the main advantages of geometric and approximation methods, a new
direction-approximation kinematic geometry of mechanism synthesis was formulated.
It studies a special class of approximation problems related to the definition of points and
lines of a rigid body describing the constraint of the synthesizing kinematic chains. In the
works [
24
,
25
], the basics of approximation kinematic geometry of the plane and spatial
movements are presented, where circular square points [
24
] and points with approximately
spherical and coplanar trajectories [
25
] are defined, which correspond to binary links of
the type RR, SS and SP
k
. Further, in the works [
26
,
27
], the concept of discrete Chebyshev
approximations was introduced for the kinematic synthesis of linkage mechanisms. The-
Appl. Sci. 2021,11, 9831 3 of 17
orems characterizing the Chebyshev circle and straight line in plane motion [
26
] and the
Chebyshev sphere and plane in spatial motion [
27
], as well as iterative algorithms for
determining Chebyshev circular, spherical and other points based on minimizing the limit
values of the weighted difference, are formulated.
Many approaches for kinematic analysis and synthesis of mechanisms and manipu-
lators are based on the derivative of loop-closure equations using the vector, matrix and
screw methods [28–42]. In this case, polynomials of high degrees are obtained [43,44].
In this paper, structural-parametric synthesis and kinematic analysis of the RoboMech
class PM with two sliders is carried out on the base of modular approach, according to
which, by the given laws of motions of the input and output links, the structural scheme
and geometric parameters of links are determined from the separate simple structural
modules [
45
]. The considered PM is formed by connecting two sliders using one passive
and one negative closing kinematic chain (structural (modules) having zero and negative
DOF, respectively. This PM can be used in a cold stamping technological—line as proposed
in a case of study.
2. Structural Scheme of a Cold Stamping Technological Line
A scheme of a simple single-stream robotic cold stamping technological line [
46
] is
shown in Figure 1, where TE is the main technological equipment, IR is an industrial robot
and S is a piece-by-piece delivery store.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 18
movements are presented, where circular square points [24] and points with approxi-
mately spherical and coplanar trajectories [25] are defined, which correspond to binary
links of the type RR, SS and SPk. Further, in the works [26,27], the concept of discrete Che-
byshev approximations was introduced for the kinematic synthesis of linkage mecha-
nisms. Theorems characterizing the Chebyshev circle and straight line in plane motion
[26] and the Chebyshev sphere and plane in spatial motion [27], as well as iterative algo-
rithms for determining Chebyshev circular, spherical and other points based on minimiz-
ing the limit values of the weighted difference, are formulated.
Many approaches for kinematic analysis and synthesis of mechanisms and manipu-
lators are based on the derivative of loop-closure equations using the vector, matrix and
screw methods [28–42]. In this case, polynomials of high degrees are obtained [43,44].
In this paper, structural-parametric synthesis and kinematic analysis of the Ro-
boMech class PM with two sliders is carried out on the base of modular approach, accord-
ing to which, by the given laws of motions of the input and output links, the structural
scheme and geometric parameters of links are determined from the separate simple struc-
tural modules [45]. The considered PM is formed by connecting two sliders using one
passive and one negative closing kinematic chain (structural (modules) having zero and
negative DOF, respectively. This PM can be used in a cold stamping technological—line
as proposed in a case of study.
2. Structural Scheme of a Cold Stamping Technological Line
A scheme of a simple single-stream robotic cold stamping technological line [46] is
shown in Figure 1, where TE is the main technological equipment, IR is an industrial robot
and S is a piece-by-piece delivery store.
Figure 1. Scheme of a single-stream robotic cold stamping technological line.
This scheme is typical for technological processes with a small cycle of processing
production items on technological equipment, in particular, in cold stamping process. In
this scheme, there is no inter-operational transport system, and products (production
items) are transferred from one piece of technological equipment to another directly by
industrial robots.
The equipment of the presented scheme of the technological line operate in the fol-
lowing sequence: the first industrial robot IR1 takes a workpiece in a certain position from
the first store S1 and delivers it to the first piece of technological equipment ТE1, where the
workpiece is processed (stamped). After primary processing, the product is delivered by
the same industrial robot IR1 to the second store S2, where the position of the product is
changed for sequent processing. Then, the second industrial robot IR2 delivers the product
from the store S2 to the second piece of technological equipment TE2, where the second
processing of the product is carried out. After this processing, the product is delivered to
the store S3 by the second industrial robot IR2. Moreover, all equipment must operate in
accordance with a given cyclogram of the technological line.
Figure 1. Scheme of a single-stream robotic cold stamping technological line.
This scheme is typical for technological processes with a small cycle of processing
production items on technological equipment, in particular, in cold stamping process.
In this scheme, there is no inter-operational transport system, and products (production
items) are transferred from one piece of technological equipment to another directly by
industrial robots.
The equipment of the presented scheme of the technological line operate in the follow-
ing sequence: the first industrial robot IR
1
takes a workpiece in a certain position from the
first store S
1
and delivers it to the first piece of technological equipment TE
1
, where the
workpiece is processed (stamped). After primary processing, the product is delivered by
the same industrial robot IR
1
to the second store S
2
, where the position of the product is
changed for sequent processing. Then, the second industrial robot IR
2
delivers the product
from the store S
2
to the second piece of technological equipment TE
2
, where the second
processing of the product is carried out. After this processing, the product is delivered to
the store S
3
by the second industrial robot IR
2
. Moreover, all equipment must operate in
accordance with a given cyclogram of the technological line.
Thus, the considered technological line for processing the product with two changing
positions of the product has two main pieces of technological equipment (hydraulic presses),
I and II, four auxiliary pieces of equipment: a device III for feeding the workpiece, a device
IV for removing the product after processing and two industrial robots V and VI (Figure 2).
These devices in total have of eight DOF. It is known that the more DOF of equipment in
Appl. Sci. 2021,11, 9831 4 of 17
technological lines for mass production of typical products, the lower their productivity
and reliability.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 18
Thus, the considered technological line for processing the product with two changing
positions of the product has two main pieces of technological equipment (hydraulic
presses), I and II, four auxiliary pieces of equipment: a device III for feeding the work-
piece, a device IV for removing the product after processing and two industrial robots V
and VI (Figure 2). These devices in total have of eight DOF. It is known that the more DOF
of equipment in technological lines for mass production of typical products, the lower
their productivity and reliability.
Figure 2. Scheme of the technological line with eight DOFs.
In order to eliminate the noted disadvantages of the technological line, we reduce its
number of DOF, replacing the technological and auxiliary equipment with the RoboMech
class PMs. According to the developed principle of forming the RoboMech class PMs [7],
we combine the main technological equipment (hydraulic presses) I and II with devices
for feeding the workpieces III and removing the workpieces IV, and also combine two
industrial robots V and VI into one PM with two end-effectors.
Combination of the hydraulic presses I and II with the devices for feeding the work-
pieces III and removing the workpieces IV into PMs I’ and II’ with two sliders is carried
out by connecting the punches Q’ and Q’’ of the hydraulic cylinders I and II with the slid-
ers P’ and P’’ of the workpieces feeding and removing devices III and IV using passive
CKCs A’B’C’ and A’’B’’C’’, as well as negative CKCs D’E’ and D’’E’’, respectively (Figure
3).
Figure 2. Scheme of the technological line with eight DOFs.
In order to eliminate the noted disadvantages of the technological line, we reduce its
number of DOF, replacing the technological and auxiliary equipment with the RoboMech
class PMs. According to the developed principle of forming the RoboMech class PMs [
7
],
we combine the main technological equipment (hydraulic presses) I and II with devices
for feeding the workpieces III and removing the workpieces IV, and also combine two
industrial robots V and VI into one PM with two end-effectors.
Combination of the hydraulic presses I and II with the devices for feeding the work-
pieces III and removing the workpieces IV into PMs I’ and II’ with two sliders is carried out
by connecting the punches Q’ and Q” of the hydraulic cylinders I and II with the sliders
P’ and P” of the workpieces feeding and removing devices III and IV using passive CKCs
A’B’C’ and A”B”C”, as well as negative CKCs D’E’ and D”E”, respectively (Figure 3).
Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 18
Figure 3. Structural scheme of the technological line with the RoboMech class PMs.
Combination of two industrial robots (serial manipulators) AIIIBIIICIII and AIVBIVCIV
into one PM III′ with two output points is carried out by connecting the links BIIICIII and
BIVCIV of the serial manipulators AIIIBIIICIII and AIV BIVCIV using negative CKC DIIIEIIIFIII.
As a result, we obtain a scheme of a technological line with the RoboMech class PMs
with three DOFs, where the presses I’ and II’ have two DOF, the PM with two output
point III’ have one DOFs. Hydraulic presses I’ and II’ with devices for feeding and remov-
ing workpieces work in the OXY plane, and the PM with two output points III’ works in
the OXZ plane. Figure 3 also shows a scheme of the PM IV′ operating in a cylindrical
coordinate system. This PM is used to store finished products in bins.
The considered technological line with the RoboMech class PMs operates as follows.
When feeding the workpiece for processing by the press I’, the slider P’ takes the right
extreme position, and the punch Q’ of the hydraulic cylinder takes the upper extreme
position. When processing the workpiece, the punch Q’ of this press takes the lower ex-
treme position, and the slider P’ returns to the left extreme position to deliver the next
workpiece. At the moment of return of the punch Q’ of the hydraulic cylinder I’ to the
upper position, the first gripper C’’’ of the PM with two output points takes the extreme
left position, captures the processed workpiece and delivers it to the store. At this mo-
ment, the second gripper CIV delivers the previously processed workpiece to the press II′
for further processing, i.e., takes the extreme upper position. After the secondary pro-
cessing of the workpiece, the finished product is delivered to the container by the slider
P’’. Then the cycle is repeated.
After accumulation of products in the container, it is stored in bins with the help of
PM IV’. A gripper СV of this PM reproduces the series of horizontal and vertical trajecto-
ries. In this case, the series of horizontal trajectories are reproduced by input link DVEV
and the series of vertical trajectories are reproduced by input link IVHV drive. Rotation of
the entire PM around the vertical axis provides a spatial movement of the gripper CV in a
cylindrical coordinate system. Structural-parametric synthesis of this PM is considered in
[47]. Let us consider the structural-parametric synthesis of the PM with two sliders.
Figure 3. Structural scheme of the technological line with the RoboMech class PMs.
Appl. Sci. 2021,11, 9831 5 of 17
Combination of two industrial robots (serial manipulators) A
III
B
III
C
III
and A
IV
B
IV
C
IV
into one PM III
0
with two output points is carried out by connecting the links B
III
C
III
and
B
IV
C
IV
of the serial manipulators A
III
B
III
C
III
and A
IV
B
IV
C
IV
using negative CKC D
III
E
III
F
III
.
As a result, we obtain a scheme of a technological line with the RoboMech class PMs
with three DOFs, where the presses I’ and II’ have two DOF, the PM with two output point
III’ have one DOFs. Hydraulic presses I’ and II’ with devices for feeding and removing
workpieces work in the OXY plane, and the PM with two output points III’ works in
the OXZ plane. Figure 3also shows a scheme of the PM IV
0
operating in a cylindrical
coordinate system. This PM is used to store finished products in bins.
The considered technological line with the RoboMech class PMs operates as follows.
When feeding the workpiece for processing by the press I’, the slider P’ takes the right
extreme position, and the punch Q’ of the hydraulic cylinder takes the upper extreme
position. When processing the workpiece, the punch Q’ of this press takes the lower
extreme position, and the slider P’ returns to the left extreme position to deliver the next
workpiece. At the moment of return of the punch Q’ of the hydraulic cylinder I’ to the
upper position, the first gripper C”’ of the PM with two output points takes the extreme
left position, captures the processed workpiece and delivers it to the store. At this moment,
the second gripper C
IV
delivers the previously processed workpiece to the press II
0
for
further processing, i.e., takes the extreme upper position. After the secondary processing of
the workpiece, the finished product is delivered to the container by the slider P”. Then the
cycle is repeated.
After accumulation of products in the container, it is stored in bins with the help of
PM IV’. A gripper C
V
of this PM reproduces the series of horizontal and vertical trajectories.
In this case, the series of horizontal trajectories are reproduced by input link D
V
E
V
and the
series of vertical trajectories are reproduced by input link I
V
H
V
drive. Rotation of the entire
PM around the vertical axis provides a spatial movement of the gripper C
V
in a cylindrical
coordinate system. Structural-parametric synthesis of this PM is considered in [
47
]. Let us
consider the structural-parametric synthesis of the PM with two sliders.
3. Structural-Parametric Synthesis of the PM with Two Sliders
The problem of structural-parametric synthesis of the PM with two sliders is to
determine the structural scheme and geometric parameters of links, when the first slider
Q (the punch of a hydraulic press) takes the lower extreme position with a stroke
sQ1
,
the second slider P takes the left extreme position with a stroke
sP1
(Figure 4a) and also,
when the first slider Q takes the upper extreme position with a stroke
sPN
, and the second
slider P takes the right extreme position with a stroke
sQN
(Figure 4b). For the convenience
of reporting the strokes of the sliders, the absolute coordinate system OXY is located at the
point of their intersection.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 18
3. Structural-Parametric Synthesis of the PM with Two Sliders
The problem of structural-parametric synthesis of the PM with two sliders is to de-
termine the structural scheme and geometric parameters of links, when the first slider Q
(the punch of a hydraulic press) takes the lower extreme position with a stroke 1
Q
s
, the
second slider P takes the left extreme position with a stroke 1
P
s
(Figure 4a) and also,
when the first slider Q takes the upper extreme position with a stroke N
P
s
, and the second
slider P takes the right extreme position with a stroke N
Q
s
(Figure 4b). For the conven-
ience of reporting the strokes of the sliders, the absolute coordinate system OXY is located
at the point of their intersection.
(a) (b)
Figure 4. (a) lower extreme and (b) upper extreme positions of the slider Q.
As noted above, to form the PM with two sliders, providing their specified positions,
we connect the punch Q of the hydraulic press and the slider P using the dyad ABC with
revolute kinematic pairs. The dyad ABC has zero DOF and it is a passive CKC, which does
not impose geometric constraints on the movements of the punch Q and the slider P.
Therefore, the passive CKC ABC allows reproduction of the specified movements of the
sliders Q and P. Then we connect the link BC of the dyad ABC with a base using a binary
link DE with revolute kinematic pairs, which has one negative DOF, it is a negative CKC.
Negative CKC DE imposes one geometric constraint on the movements of the sliders Q
and P, and as a result, we obtain a structural scheme of the PM with structural formula I
(0,1) → III (3,4,2,5), where the kinematic chain 3-4-2-5 represents the Assur group of the
third class [48]. Figure 5 shows a block structure of the formed PM with two sliders.
Figure 5. Block structure of the PM with two sliders.
For parametric synthesis of the PM with two sliders, let us consider its i-th interme-
diate position and attach the coordinate systems 11 2 2
an d Q
х
yPхy with the sliders (Figure
Figure 4. (a) lower extreme and (b) upper extreme positions of the slider Q.
Appl. Sci. 2021,11, 9831 6 of 17
As noted above, to form the PM with two sliders, providing their specified positions,
we connect the punch Q of the hydraulic press and the slider P using the dyad ABC with
revolute kinematic pairs. The dyad ABC has zero DOF and it is a passive CKC, which
does not impose geometric constraints on the movements of the punch Q and the slider P.
Therefore, the passive CKC ABC allows reproduction of the specified movements of the
sliders Q and P. Then we connect the link BC of the dyad ABC with a base using a binary
link DE with revolute kinematic pairs, which has one negative DOF, it is a negative CKC.
Negative CKC DE imposes one geometric constraint on the movements of the sliders Q
and P, and as a result, we obtain a structural scheme of the PM with structural formula
I (0,1)
→
III (3,4,2,5), where the kinematic chain 3-4-2-5 represents the Assur group of the
third class [48]. Figure 5shows a block structure of the formed PM with two sliders.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 18
3. Structural-Parametric Synthesis of the PM with Two Sliders
The problem of structural-parametric synthesis of the PM with two sliders is to de-
termine the structural scheme and geometric parameters of links, when the first slider Q
(the punch of a hydraulic press) takes the lower extreme position with a stroke 1
Q
s
, the
second slider P takes the left extreme position with a stroke 1
P
s
(Figure 4a) and also,
when the first slider Q takes the upper extreme position with a stroke N
P
s
, and the second
slider P takes the right extreme position with a stroke N
Q
s
(Figure 4b). For the conven-
ience of reporting the strokes of the sliders, the absolute coordinate system OXY is located
at the point of their intersection.
(a) (b)
Figure 4. (a) lower extreme and (b) upper extreme positions of the slider Q.
As noted above, to form the PM with two sliders, providing their specified positions,
we connect the punch Q of the hydraulic press and the slider P using the dyad ABC with
revolute kinematic pairs. The dyad ABC has zero DOF and it is a passive CKC, which does
not impose geometric constraints on the movements of the punch Q and the slider P.
Therefore, the passive CKC ABC allows reproduction of the specified movements of the
sliders Q and P. Then we connect the link BC of the dyad ABC with a base using a binary
link DE with revolute kinematic pairs, which has one negative DOF, it is a negative CKC.
Negative CKC DE imposes one geometric constraint on the movements of the sliders Q
and P, and as a result, we obtain a structural scheme of the PM with structural formula I
(0,1) → III (3,4,2,5), where the kinematic chain 3-4-2-5 represents the Assur group of the
third class [48]. Figure 5 shows a block structure of the formed PM with two sliders.
Figure 5. Block structure of the PM with two sliders.
For parametric synthesis of the PM with two sliders, let us consider its i-th interme-
diate position and attach the coordinate systems 11 2 2
an d Q
х
yPхy with the sliders (Figure
Figure 5. Block structure of the PM with two sliders.
For parametric synthesis of the PM with two sliders, let us consider its i-th intermediate
position and attach the coordinate systems
Qx1y1
and
Px2y2
with the sliders (Figure 6),
the axes
Px1
and
Qx2
which are directed parallel to the axis OX of the absolute coordinate
system OXY. Then, the movements of the sliders are determined by the parameters
sQi
and
sPi
of the coordinate systems
Px2y2
and Qx
2
y
2
movements, where i= 1, 2 ..., N(Nis the
number of given positions). Parametric synthesis of this PM with two sliders, according to
its block structure (Figure 5), consists of a parametric synthesis of the passive CKC ABC
and the negative CKC DE.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 18
6), the axes 12
an d P
х
Qх which are directed parallel to the axis OX of the absolute coordi-
nate system OXY. Then, the movements of the sliders are determined by the parameters
i
Q
s
and i
P
s
of the coordinate systems 22
P
х
y and Qх2y2 movements, where i = 1, 2 ..., N
(N is the number of given positions). Parametric synthesis of this PM with two sliders,
according to its block structure (Figure 5), consists of a parametric synthesis of the passive
CKC ABC and the negative CKC DE.
Figure 6. Intermediate position of the PM with two sliders.
The synthesis parameters (geometric parameters of the links) of the passive CKC
ABC is a vector р=𝑥(),𝑦(),𝑥(),𝑦(),𝑙,𝑙В, where 𝑥(),𝑦() and 𝑥(),𝑦() are the
coordinates of the joints A and C in the moving coordinate systems𝑄𝑥𝑦 and 𝑃𝑥𝑦
respectively, 𝑙 and 𝑙В are the lengths of the links AB and BC. Since the passive CKC
ABC does not impose geometric constraint on the movements of the sliders Q and P,
then its synthesis parameters are set, they are varied by the generator of 𝐿𝑃 sequence
[49] depending on the geometric constraint imposed by the negative CKC DE. Negative
CKC DE imposes one geometric constraint on the movements of the links AB and BC of
the passive CKC ABC; therefore, its synthesis parameters are determined.
Let us consider the parametric synthesis of the negative CKC DE. To do this, it is
necessary to first determine the angle φ4i by the expression
𝜑=𝜑(СА)+𝑐𝑜𝑠𝑙ВС
+𝑙(СА)
−𝑙АВ
𝑙ВС𝑙(СА), (1)
where
𝜑(СА)=tg 𝑌А−𝑌С
𝑋А−𝑋С, (2)
𝑙(СА)=[(𝑋А−𝑋)+(𝑌−𝑌)]
. (3)
Coordinates 𝑋,𝑌 and 𝑋,𝑌 of the joints A and C in the absolute coordinate sys-
tem OXY in Equations (2) and (3) are determined by the expressions
𝑋=𝑥(),𝑌А=𝑠+𝑦(),𝑋=−𝑠+𝑥(),𝑌 =𝑦(). (4)
With the link CB of the dyad ABC, we attach the coordinate system Сх𝑦, the axis
Сх of which is directed along the link CB. Then, the synthesis parameters of the negative
CKC DE are the vector р=𝑥Е(),𝑦Е(),𝑋,𝑌,𝑙, where 𝑥Е(),𝑦Е() and 𝑋,𝑌 are the
coordinates of the joints E and D in the coordinate systems Сх𝑦 and OXY, respectively.
Figure 6. Intermediate position of the PM with two sliders.
The synthesis parameters (geometric parameters of the links) of the passive CKC
ABC is a vector
r1=hx(1)
A,y(1)
A,x(2)
C,y(2)
C,lAB,lBCiT
, where
x(1)
A
,
y(1)
A
and
x(2)
C
,
y(2)
C
are the
coordinates of the joints A and C in the moving coordinate systems
Qx1y1
and
Px2y2
respectively,
lAB
and
lBC
are the lengths of the links AB and BC. Since the passive CKC
ABC does not impose geometric constraint on the movements of the sliders Q and P,
then its synthesis parameters are set, they are varied by the generator of
LPτ
sequence [
49
]
Appl. Sci. 2021,11, 9831 7 of 17
depending on the geometric constraint imposed by the negative CKC DE. Negative CKC
DE imposes one geometric constraint on the movements of the links AB and BC of the
passive CKC ABC; therefore, its synthesis parameters are determined.
Let us consider the parametric synthesis of the negative CKC DE. To do this, it is
necessary to first determine the angle ϕ4iby the expression
ϕ4i=ϕ(CA)i+cos−1l2
BC +l2
(CA)i−l2
AB
lBCl(CA)i
, (1)
where
ϕ(CA)i=tg−1YAi−YCi
XAi−XCi
, (2)
l(CA)i= [(XAi−XCi)2+ (YAi−YCi)2]
1
2. (3)
Coordinates
XAi
,
YAi
and
XCi
,
YCi
of the joints Aand Cin the absolute coordinate
system OXY in Equations (2) and (3) are determined by the expressions
XAi=x(1)
A,YAi=sQi+y(1)
A,XCi=−sPi+x(2)
C,YCi =y(2)
C. (4)
With the link CB of the dyad ABC, we attach the coordinate system
Cx4y4
, the axis
Cx4
of which is directed along the link CB. Then, the synthesis parameters of the negative
CKC DE are the vector
r2=hx(4)
E,y(4)
E,XD,YD,lDEiT
, where
x(4)
E
,
y(4)
E
and
XD
,
YD
are the
coordinates of the joints Eand Din the coordinate systems Cx4y4and OXY, respectively.
Let us consider the motion of the plane
Cx4y4
in the absolute coordinate system
OXY. In this case, the joint
Ex(4)
E,y(4)
E
moves along an arc of a circle with a center at the
joint
D(XD,YD)
and a radius
lDE
. Consequently, an equation of the geometric constraint
imposed by the negative CKC DE of the RR type on the motion of the moving plane
Cx4y4
is expressed as a weighted difference
∆qi=XEi−XD2+YEi−YD2−l2
DE , (5)
where
XEi
and
YEi
are the coordinates of the joint Ein the absolute coordinate system OXY,
which are determined by the expression
XEi
YEi=XCi
YCi+cosϕ4i−sinϕ4i
sinϕ4icosϕ4i·"x(2)
E
y(2)
E#(6)
After substituting Equation (6) into Equation (5) and replacing the synthesis parame-
ters of the form
p1
p2=XD
YD,p4
p5="x(4)
E
y(4)
E#,p3=1
2X2
D+Y2
D+x(4)2
E+y(4)2
E−l2
ED (7)
then Equation (5) is expressed linearly in two groups of synthesis parameters
p(1)
2=[p1,p2,p3]T
and p(2)
2=[p4,p5,p3]Tin the form
∆q(k)
ip(k)
2=2g(k)T
i·p(k)
2−g(k)
oi ,k=1, 2, (8)
Appl. Sci. 2021,11, 9831 8 of 17
where
g(k)
i
and
g(k)
oi
are the coefficients of the vectors
p(k)
2
and free terms depending on the
remaining synthesis parameters, which have the form
g(1)
i=−
XCi
YCi
1
−
Γ(ϕ4i)0
0
0 0 1
·
p4
p5
0
(9)
g(2)
i=
Γ(ϕ4i)0
0
0 0 1
·
XCi
YCi
1
+
Γ(ϕ4i)0
0
0 0 1
·
p1
p2
0
(10)
g(1)
oi =−1
2hX2
Ci+Y2
Cii+XCi,YCi·Γ(ϕ4i)p4
p5(11)
g(2)
oi =−1
2hX2
Ci+Y2
Cii−XCi,YCi·p1
p2, (12)
where Γ(α)is a rotation matrix of view
Γ(α)=cosα−sinα
sinαcosα. (13)
The linear representability of Equation (5) allows one to formulate and solve the
Chebyshev and least-square approximations for parametric synthesis [
50
]. In the Cheby-
shev approximation problem, the vectors of synthesis parameters are determined from the
minimum of the functional
S(k)p(k)
2=max
i=1,N∆q(k)
ij p(k)
22→min
p(k)
2
S(k)p(k)
2(14)
In the least-square approximation problem, the synthesis parameters vectors are
determined from the minimum of the functional
S(k)p(k)
2=
M
∑
i=1∆q(k)
ip(k)
22→min
p(k)
2
S(k)p(k)
2. (15)
The linear representability of Equation (5) allows the use of the kinematic inversion
method, which is an iterative process, at each step of which one group of synthesis param-
eters
p(k)
2
is determined to solve the Chebyshev approximation problem (Equation (14)).
In this case, the linear programming problem is solved [
51
]. To do this, we introduce a
new variable
p0=ε
, where
ε
is the required approximation accuracy. Then the minimax
problem (Equation (14)) leads to the following linear programming problem: determine
the minimum of the sum
σ=cT·x→min
xσ, (16)
with the following constraints
h0T
i·x+h0i≥0,h00T·x−h0i≥0, (17)
where
c=[0, . . . , 0, 1]T,x=hp(k)
2,p0iT,h0
i=h−g(k)
i, 0.5iT, (18)
h00 =hg(k)
i, 0.5iT,h0i=g0i. (19)
The sequence of the obtained values of the function
S(k)
will decrease and have a limit
as a sequence bounded below, because S(k)p(k)
2≥0 for any p(k)
2.
Appl. Sci. 2021,11, 9831 9 of 17
Let consider the least-square approximation problem (Equation (15)) for the synthesis.
The necessary conditions for the minimum of functions (Equation (15)) with respect to the
parameters p(k)
2
∂S(k)
i
∂p(k)
2
=0, (20)
leading to the systems of linear equations
H(k)·p(k)
2=h(k), (21)
where
H(k)=
N
∑
i=1
g(k)
1i2g(k)
1ig(k)
2ig(k)
1i2
g(k)
1i·g(k)
2ig(k)
2i2g(k)
2i2
g1ig(k)
2i21
, (22)
h(k)=
N
∑
i=1
g(k)
1i·g(k)
0i
g(k)
2i·g(k)
0i
g0i
. (23)
Solving these systems of linear equations for each group of synthesis parameters for
given values of the remaining groups of synthesis parameters, we determine their values
r(k)
2=H(k)−1·h(k), (24)
at
det(H(k))6=
0. If
det(H(k)) =
0, then the revolute kinematic pair is replaced by prismatic
kinematic pair.
The matrix
H(k)
can be represented as a product
H1·H1T
, where
H1
is a matrix with
dimension r×N(in the considered case r=3)
H1=
g(k)
11 g(k)
12 . . . g(k)
1N
g(k)
21 g(k)
22 . . . g(k)
2N
.
.
..
.
. . . . .
.
.
g(k)
r1g(k)
r2. . . g(k)
rN
. (25)
According to the Binet-Cauchy formula [
52
], the determinant
detHk
becomes into the
sum of the squares of all minors
Hαβγ
of order r(we assume that
N≥r
) in the matrix
H1
,
compiled in ascending order of the column indices, i.e.,
detHk=∑
1≤α≤β≤γ
H2
αβγ . (26)
Consequently, the determinant
detHk
is positive definite together with the principal
minors and the solution of the set of linear Equation (21) corresponds to the minimum of
the function Swith respect to the parameters
p(k)
. Hence, the least-square approximation
problem (Equation (15)) can be solved by the linear iterations method, at each step of which
one group of parameters
p(k)
is determined. The sequence of values of the function Swill
be decreasing and have a limit as a sequence bounded below.
4. Kinematic Analysis of the PM with Two Sliders
Given the synthesis parameters and the positions
sQi
of the hydraulic cylinder punch
Q, it is necessary to find the kinematic parameters of the slider P.
This PM (Figure 6) has the structural formula
I(1)→III(2, 3, 4, 5), (27)
Appl. Sci. 2021,11, 9831 10 of 17
i.e., the PM contains an Assur group of the third class with one external prismatic kinematic
pair [41]. In the literature, there is no solution of kinematics of this type group.
4.1. Position Analysis
For position analysis of the considered PM, we use the method of conditional general-
ized coordinates [
50
]. According to this method, we remove the link 5 by disconnecting
the elements of the joints Dand Eand select the slider Pas a conditional input link due
to the additional DOF that appears. Then, this PM of the third class is transformed into a
mechanism of the second class with the structural formula
I(1)→II(3, 4)←I(2). (28)
Derive the function
∆=lDE −e
lDE , (29)
where
e
lDE
is a variable distance between the centers of the disconnected joints Dand E,
which is determined by the expression
e
lDE =hKhDi−KhE2+YDi−YE2i1
2. (30)
Coordinates
KhDi
and
YDi
of the joint Dcenter in Equation (30) are determined by
the equation
"XDi
YCi#="XCi
YCi#+cosϕ4i−sinϕ4i
sinϕ4icosϕ4i·"x(4)
D
y(4)
D#, (31)
where XCi
YCi="−sPi+x(2)
C
y(2)
C#. (32)
To determine the angle
ϕ4i
in Equation (31), we derive a vector ABC loop-closure equation
lABe3i−lCDBe4i+l(CA)ie(CA)i=0, (33)
l(CA)i= [(XAi−XCi)2+ (YAi−YCi)2]
1
2, (34)
ϕ(CA)i=tg−1YAi−YCi
XAi−XCi
, (35)
XAi
YAi="−x(1)
A
sQi+y(1)
A#. (36)
In Equation (33) edenotes the unit vector.
We transfer
lABe3i
to the right side of Equation (33) and square both sides. As a result,
we obtain
ϕ4i=ϕ(CA)i+cos−1l2
CB +l2
(CA)i−l2
AB
2lCB l(CA)i
, (37)
Next, we define XBi
YBi=XCi
YCi+lCB cos ϕ4i
sinϕ4i, (38)
ϕ3i=tg−1YBi−YAi
XBi−XAi
. (39)
Thus, Equation (29) is a function of one variable: the conditional generalized co-
ordinate
sPi
, for the given values of the real generalized coordinate
sQi
. Minimizing
Appl. Sci. 2021,11, 9831 11 of 17
Equation (29) with respect to a variable
sPi
by the bisection method [
53
], we determine
its values for given values
sQi
. In this case, the angles
ϕ3i
and
ϕ4i
are simultaneously
determined. The angle ϕ5iis determined by the expression
ϕ5i=tg−1YDi−YE
XDi−XE
. (40)
4.2. Analogues of Velocities and Accelerations
To solve the problems of analogues of velocities and accelerations of the PM with two
sliders, we select its independent vector contours, the number of which is equal to half the
number of links of the Assur group, i.e., it is equal to two. As independent vector contours,
we choose the contours OQ’ABCC’O and OQ’ABDEO, the vector loop-closure equations of
which have the forms
lOQ0eOQ0+lQ0AeQ0A+lABe3i−lC Be4i+lCC0eCC0−l(OC0)ieOC0=0
lOQ0eOQ0+lQ0AeQ0A+lABe3i−lD Be(D B)i−lED e5i−lOE eOE =0)(41)
Project the system of Equation (41) on the axes OX and OY of the absolute coordinate
system OXY
lQ0A+lABcosϕ3i−lC Bcosϕ4i−l(OC0)i=0
lOQ0+lABsinϕ3i−lCB sinϕ4i+lCC0=0
lQ0A+lABcosϕ3i−lD Bcos(ϕ4i−α4)−lE Dcos ϕ5i−lOE cosϕOE =0
lOQ0+lABsinϕ3i−lDB sin(ϕ4i−α4)−lED sinϕ5i−lOEsin ϕOE =0
(42)
Differentiate the system of Equation (42) with respect to the generalized coordinate
sQi
−lABsinϕ3i·ϕ0
3i−lCB sinϕ4i·ϕ0
4i−uPi=0
1+lABcosϕ3i·ϕ0
3i−lCB cosϕ4i·ϕ0
4i=0
−lABsinϕ3i·ϕ0
3i+lDB sin(ϕ4i−α4)·ϕ0
4i+lED sinϕ5i·ϕ0
5i=0
1+lABcosϕ3i·ϕ0
3i−lDB cos(ϕ4i−α4)·ϕ0
4i−lED cosϕ5i·ϕ0
5i=0
(43)
From the system of Equation (43) we determine the analogues of velocities
u=A−1·b, (44)
where
A=
YAi−YBiYBi−YCi0−1
XBi−XAiXCi−XBi0 0
YAi−YBiYBi−YDiYDi−YE0
XBi−XAiXDi−XBiXE−XDi0
(45)
u=
ϕ0
3i
ϕ0
4i
ϕ0
5i
uPi
,b=
0
−1
0
0
. (46)
Differentiate the system of Equation (43) with respect to the generalized coordinate
sQi
−lABcosϕ3i·ϕ023i−lABsinϕ3i·ϕ00
3i+lCB cosϕ4i·ϕ02
4i+lCB sinϕ4i·ϕ0 0
4i−w00
Pi=0
−lABsinϕ3i·ϕ02
3i+lABcosϕ3i·ϕ00
3i+lCB sinϕ4i·ϕ02
4i+lCB cosϕ4i·ϕ0 0
4i=0
−lABcosϕ3i·ϕ023i−lABsinϕ3i·ϕ00
3i+lDB cos(ϕ4i−α4)·ϕ02
4i+
+lDB sin(ϕ4i−α4)·ϕ00
4i+lED cosϕ5i·ϕ02
5i+lED sinϕ5i·ϕ0 0
5i=0
−lABsinϕ3i·ϕ023i−lABcosϕ3i·ϕ00
3i+lDB sin(ϕ4i−α4)·ϕ02
4i−
−lDB cos(ϕ4i−α4)·ϕ00
4i+lED sinϕ5i·ϕ02
5i−lED cosϕ5i·ϕ0 0
5i=0
(47)
Appl. Sci. 2021,11, 9831 12 of 17
Then we obtain the analogues of accelerations
w=A−1·c, (48)
where
w=
ϕ00
3i
ϕ00
4i
ϕ00
5i
wPi
,c=
XBi−XAi·ϕ02
3i+XCi−XBi·ϕ02
4i
YBi−YAi·ϕ02
3i+YCi−YBi·ϕ02
4i
XBi−XAi·ϕ02
3i+XDi−XBi·ϕ02
4i+XEi−XDi·ϕ02
5i
YBi−YAi·ϕ02
3i+YDi−YBi·ϕ02
4i+YEi−YDi·ϕ02
5i
(49)
5. Numerical Results and Prototyping
N= 11 positions
sQi
and
sPi
of the input and output sliders of the PM with two sliders
are shown in Table 1.
Table 1. Positions of the input and output sliders.
i 1 2 3 4 5 6 7 8 9 10 11
sQi, mm 0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60
sPi, mm −97.88 −87.86 −80.29 −73.46 −66.8 −59.97 −52.63 −44.31 −34.18 −19.99 0
Tables 2and 3show the obtained values of the synthesis parameters of the passive
CKC ABC and negative CKC DE, respectively.
Table 2. Synthesis parameters of the passive CKC ABC.
x(1)
A,mm y(1)
A,mm x(2)
C,mm y(2)
C,mm lAB,mm lBC ,mm
−7.5012 2.0817 −2.5335 1.7562 60.0174 100.0207
Table 3. Synthesis parameters of the negative CKC DE.
x(4)
D,mm y(4)
D,mm XЕ,mm YЕ,mm lED,mm
50.0628 −20.0408 −69.5361 67.9353 60.7365
3D CAD model of the synthesised PM with two sliders is shown in Figure 7.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 18
Then we obtain the analogues of accelerations
𝒘=
𝑨
⋅𝒄, (48)
where
𝒘=⎣
⎢
⎢
⎢
⎡
𝜑
″
𝜑
″
𝜑
″
𝑤⎦
⎥
⎥
⎥
⎤
,𝒄=⎣
⎢
⎢
⎢
⎢
⎢
⎡
𝑋−𝑋⋅𝜑′
+𝑋−𝑋⋅𝜑′
𝑌−𝑌⋅𝜑′
+𝑌−𝑌⋅𝜑′
𝑋−𝑋⋅𝜑′
+𝑋−𝑋⋅𝜑′
+𝑋−𝑋⋅𝜑′
𝑌−𝑌⋅𝜑′
+𝑌−𝑌⋅𝜑′
+𝑌−𝑌⋅𝜑′
⎦
⎥
⎥
⎥
⎥
⎥
⎤
(49)
5. Numerical Results and Prototyping
N = 11 positions 𝒔𝑸𝒊 and 𝒔𝑷𝒊 of the input and output sliders of the PM with two slid-
ers are shown in Table 1.
Table 1. Positions of the input and output sliders.
i 1 2 3 4 5 6 7 8 9 10 11
𝑠, mm 0 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 60
𝑠, mm −97.88 −87.86 −80.29 −73.46 −66.8 −59.97 −52.63 −44.31 −34.18 −19.99 0
Tables 2 and 3 show the obtained values of the synthesis parameters of the passive
CКC ABC and negative CKC DE, respectively.
Table 2. Synthesis parameters of the passive CKC ABC.
𝒙𝑨
(𝟏),mm 𝒚𝑨
(𝟏),mm 𝒙𝑪
(𝟐),mm 𝒚𝑪
(𝟐),mm 𝒍𝑨𝑩,mm 𝒍𝑩𝑪,mm
−7.5012 2.0817 −2.5335 1.7562 60.0174 100.0207
Table 3. Synthesis parameters of the negative CKC DE.
𝒙𝑫
(𝟒),mm 𝒚𝑫
(𝟒),mm
𝑿
Е,mm 𝒀Е,mm 𝒍𝑬𝑫,mm
50.0628 −20.0408 −69.5361 67.9353 60.7365
3D CAD model of the synthesised PM with two sliders is shown in Figure 7.
Figure 7. 3D CAD model of the PM with two sliders.
Figure 7. 3D CAD model of the PM with two sliders.
Appl. Sci. 2021,11, 9831 13 of 17
Table 4shows the obtained values of the positions
sPi
and analogues of the linear
velocities uPiand linear accelerations wPiof the output slider P.
Table 4. Positions and anologues of the linear velocities and accelerations of the output slider P.
i 1 2 3 4 5 6 7 8 9 10 11
sPi, mm −97.88 −87.86 −
80.29
−73.46 −66.8 −59.97 −52.63 −44.31 −34.18 −19.99 0
uPi, 29.0364 24.8392
20.6181
16.0497 10.9754 5.2438 −1.3316 −9.0244 −
18.3301
−
30.4705
−
44.6353
wPi
,
mm−11.8788 1.4345 1.1417 08.1868 07.3624 1.02159 1.4427 2.2557 3.5165 3.6692 1.9001
Graphics of the parameters sPi,uPi,wPiare shown in Figure 8.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 18
Table 4 shows the obtained values of the positions 𝑠 and analogues of the linear
velocities 𝑢 and linear accelerations 𝑤 of the output slider P.
Table 4. Positions and anologues of the linear velocities and accelerations of the output slider P.
i 1 2 3 4 5 6 7 8 9 10 11
𝑠, mm −97.88 −87.86 −80.29 −73.46 −66.8 −59.97 −52.63 −44.31 −34.18 −19.99 0
𝑢, 29.0364 24.8392 20.6181 16.0497 10.9754 5.2438 −1.3316 −9.0244 −18.3301 −30.4705 −44.6353
𝑤, 𝑚𝑚 1.8788 1.4345 1.1417 08.1868 07.3624 1.02159 1.4427 2.2557 3.5165 3.6692 1.9001
Graphics of the parameters 𝑠, 𝑢, 𝑤 are shown in Figure 8.
Figure 8. Graphics of the parameters 𝑠,𝑢,𝑤.
A prototype of the PM with two sliders, and a block scheme of its characteristics are
shown in Figures 9 and 10, respectively.
(a) (b)
Figure 9. Prototype of the PM with two sliders: (a) the first position (b) the second position.
Figure 8. Graphics of the parameters sPi,uPi,wPi.
A prototype of the PM with two sliders, and a block scheme of its characteristics are
shown in Figures 9and 10, respectively.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 18
Table 4 shows the obtained values of the positions 𝑠 and analogues of the linear
velocities 𝑢 and linear accelerations 𝑤 of the output slider P.
Table 4. Positions and anologues of the linear velocities and accelerations of the output slider P.
i 1 2 3 4 5 6 7 8 9 10 11
𝑠, mm −97.88 −87.86 −80.29 −73.46 −66.8 −59.97 −52.63 −44.31 −34.18 −19.99 0
𝑢, 29.0364 24.8392 20.6181 16.0497 10.9754 5.2438 −1.3316 −9.0244 −18.3301 −30.4705 −44.6353
𝑤, 𝑚𝑚 1.8788 1.4345 1.1417 08.1868 07.3624 1.02159 1.4427 2.2557 3.5165 3.6692 1.9001
Graphics of the parameters 𝑠, 𝑢, 𝑤 are shown in Figure 8.
Figure 8. Graphics of the parameters 𝑠,𝑢,𝑤.
A prototype of the PM with two sliders, and a block scheme of its characteristics are
shown in Figures 9 and 10, respectively.
(a) (b)
Figure 9. Prototype of the PM with two sliders: (a) the first position (b) the second position.
Figure 9. Prototype of the PM with two sliders: (a) the first position (b) the second position.
Appl. Sci. 2021,11, 9831 14 of 17
Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 18
Figure 10. Block scheme of the PM with two sliders work.
At the beginning, the hydraulic cylinder punch is located at the upper extreme posi-
tion, and the distance sensor 1 checks a presence of the workpiece for stamping in the die.
If there is no workpiece, then the presence of the workpiece in the store is checked using
the distance sensor 2. If there is no workpiece in the store, the motor does not turn on. If
there is the workpiece in the store, the motor turns on and the punch moves to the lower
working position. At this time, the PM slider moves to the left extreme position for the
next workpiece.
The first stroke of the punch will be idle. After reaching the lower extreme position
of the punch, the distance sensor 1 gives the command to the hydraulic cylinder valve to
switch and the punch rises until reaching the touch sensor. In this case, the hydraulic cyl-
inder valve switches, the hydraulic cylinder motor is turned off, and the PM delivers the
workpiece to the die (working area). Further, the distance sensor 1 checks the presence of
the workpiece in the die.
After delivering the workpiece to the die, the motor turns on and the punch goes
down to the lower working position, where the weight sensor is located, which regulates
the press force for high-quality stamping. The hydraulic cylinder valve switches after
Figure 10. Block scheme of the PM with two sliders work.
At the beginning, the hydraulic cylinder punch is located at the upper extreme position,
and the distance sensor 1 checks a presence of the workpiece for stamping in the die.
If there is no workpiece, then the presence of the workpiece in the store is checked using
the distance sensor 2. If there is no workpiece in the store, the motor does not turn on.
If there is the workpiece in the store, the motor turns on and the punch moves to the lower
working position. At this time, the PM slider moves to the left extreme position for the
next workpiece.
The first stroke of the punch will be idle. After reaching the lower extreme position
of the punch, the distance sensor 1 gives the command to the hydraulic cylinder valve
to switch and the punch rises until reaching the touch sensor. In this case, the hydraulic
cylinder valve switches, the hydraulic cylinder motor is turned off, and the PM delivers
the workpiece to the die (working area). Further, the distance sensor 1 checks the presence
of the workpiece in the die.
After delivering the workpiece to the die, the motor turns on and the punch goes
down to the lower working position, where the weight sensor is located, which regulates
Appl. Sci. 2021,11, 9831 15 of 17
the press force for high-quality stamping. The hydraulic cylinder valve switches after
punching, and the punch rises until it reaches the touch sensor. In this case, the hydraulic
cylinder valve is switched and its motor is turned off. This cycle is repeated until the end
of the workpieces in the store. The connection scheme of the sensors and motor is shown
in Figure 11.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 18
punching, and the punch rises until it reaches the touch sensor. In this case, the hydraulic
cylinder valve is switched and its motor is turned off. This cycle is repeated until the end
of the workpieces in the store. The connection scheme of the sensors and motor is shown
in Figure 11.
Figure 11. Connection scheme of the sensors and motor.
6. Conclusions
The scheme of a cold-stamping technological line with the use of RoboMech class
PMs has been developed. This technological line uses three RoboMech class PMs: a PM
with two sliders, a PM with two end-effectors, and a PM working in a cylindrical coordi-
nate system. The PM with two sliders is formed by connecting two sliders (input and out-
put objects) and a base using one passive and one negative CKC. The formed PM with
two sliders contains an Assur group of the third class with one external prismatic kine-
matic pair. Geometric parameters of the negative CKC are determined on the basis of the
Chebyshev and least-square approximations. The problem of the positions of the PM with
two sliders is solved using the method of conditional generalized coordinates. The 3D
CAD model and prototype of the PM with two sliders have been made.
Author Contributions: Z.B., M.A.L., G.C. developed methods of parametric synthesis and kine-
matic analysis, A.M., B.A., Y.Z. performed numerical calculations and prototyping of the RoboMech
class PM with two sliders. All authors have read and agreed to the published version of the manu-
script.
Funding: This research is funded by the Science Committee of the Ministry of Education and Science
of Kazakhstan (Grant No AP08857522).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in the study are available on request from the
corresponding author.
Figure 11. Connection scheme of the sensors and motor.
6. Conclusions
The scheme of a cold-stamping technological line with the use of RoboMech class PMs
has been developed. This technological line uses three RoboMech class PMs: a PM with
two sliders, a PM with two end-effectors, and a PM working in a cylindrical coordinate
system. The PM with two sliders is formed by connecting two sliders (input and output
objects) and a base using one passive and one negative CKC. The formed PM with two
sliders contains an Assur group of the third class with one external prismatic kinematic pair.
Geometric parameters of the negative CKC are determined on the basis of the Chebyshev
and least-square approximations. The problem of the positions of the PM with two sliders
is solved using the method of conditional generalized coordinates. The 3D CAD model
and prototype of the PM with two sliders have been made.
Author Contributions:
Z.B., M.A.L., G.C. developed methods of parametric synthesis and kinematic
analysis, A.M., B.A., Y.Z. performed numerical calculations and prototyping of the RoboMech class
PM with two sliders. All authors have read and agreed to the published version of the manuscript.
Funding:
This research is funded by the Science Committee of the Ministry of Education and Science
of Kazakhstan (Grant No AP08857522).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The data presented in the study are available on request from the
corresponding author.
Appl. Sci. 2021,11, 9831 16 of 17
Conflicts of Interest: The authors no conflict of interest.
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