Realization and Simulation a Novel Kind of Parallel Cables-Based
Robot with Five Cables
FOUED INEL1, MOHAMMED KHADEM2, ABDELGHAFOUR SLIMANE TICH TICH3
1Mechanical Engineering Department, Automatic Laboratory, University of Skikda, ALGERIA
2Mechanical Engineering Department LGMM Laboratory, University of Skikda, ALGERIA
3Mechanical Engineering Department LGMM Laboratory, University of Skikda, ALGERIA
Abstract. In this paper, we introduce the pyramidal robot, a novel kind of parallel cable-based robot that has
been constructed and designed with five cables. Last, we suggested a control method. In this context, we
studied the application of the Runge-Kutta method of fourth order for resolving the non-linear partial
differential equations of our system, which is frequently employed for managing uncertainties in linear systems.
The primary contribution of this study is firstly the design of a reel prototype and the creation and
implementation of a graphical user interface (GUI) for displaying the position of the end effector. Second, to
test the precision of the tracking of the object, we analyse the system's response using cutting-edge methods
such as predictive control. Finally, using the advanced technique proposed, we present the simulation results on
this cable-based robot. These results demonstrate the performance of the technique as proposed.
Key Words: Dynamical Systems, Robotics, Cable based parallel robot; simulation; Predictive control; GUI.
Received: April 8, 2022. Revised: October 25, 2022. Accepted: November 21, 2022. Published: December 31, 2022.
1 Introduction
cable-based parallel robots are a unique type of
parallel manipulators robot in which the end-
effectors are powered by cables instead of rigid
links[1-3] and The movement is given by the
winding and unwinding of cables [2, 8, 9].When
compared to the robots of traditional architecture,
these provide undeniable advantages [4–7]. This
last device is a sort of parallel manipulator that
connects a fixed base to a movable platform using
cables [10, 11]. The coordinate controller of cable
lengths and tensions permit the displacement and
the application of efforts on the platform. These
robots have few moving parts, with reduced mass,
and are most suitable for tasks requiring high
performance such as speed and accuracy and
provide a large workspace. By moving the cable's
connection points, it is possible to obtain
reconfigurable manipulators. In addition, they are
easy to mount; dismount and transport, in other
hand, the main disadvantages of parallel
manipulators lie in the nature of the cables that can
only work in one direction rather than the traction
[12-14]. The best-known application is the Skycam,
a camera controlled by a cable mechanism that is
used for tele-diffusion of professional football
games. Another area of interest in biomedical
applications such as tracking the movement of
body parts. An example is the CaTraSys (Cassino
Tracking System) was used for the identification of
kinematic parameters and the mobility of man [15,
16].
The unique parallel manipulators known as cable-
based parallel robots a suitable control technique is
essential for cable-based robots in order to achieve
proper motions without damaging the cables.
According to reports like those in [17-19,20], the
Predictive approach has been created to increase
the robustness of robotic system control. An
adaptive predictive controller in particular can
adjust the control torque based on current position.
The purpose of this work is fulfilled through the
design of a new prototype pyramidal cable-based
robot with five cables, followed by the
implementation of dynamics modelling with a
predictive controller for control on real. Finally,
simulation results have been applied to verify and
assess the suggested the algorithm performs.
2 System Description
Figure 1 shows our reel prototype robot with five
cables. The base is fixed and each cable is attached
to the one end of the platform. As a result of motor
moments, cable wraps and winds the cables around
the pulley to control the position and the orientation
of the end-effector. The five cables-based robots
allow a 3D plane movement with 5 degrees of
freedom and figure 2 shows the general
geometrical axes and the vector analysis
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.30
Foued Inel, Mohammed Khadem,
Abdelghafour Slimane Tich Tich
E-ISSN: 2224-266X
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Fig 1: A Reel prototype pyramidal based parallel
robot
Fig 2: The general geometrical axes and the vector
analysis
With:
The geometric parameters that describes the robot
are listed in the following :
Li: length of the cable;
:Unitary matrix ;
R: side length of the robot base (The shape of the
robot base is square) ;
H: height between the base and the motor 5;
Mi: exit point of the cables from the base;
: vector to (a , M1);
: vector to (ò , o);
: vector to (M1, o);
: vector to (a, ò).
k: is the side length of the end-effector (The shape
of the end-effector is square).
3 Geometric Modelling
In this section, we present the inverse geometric
model for five cable-based robots.
3.1 Establishment of Mathematical Model
This model aims to determine the lengths of the
cables "Li", the angles "i" between the X,Y axes
and the cables connected to the mobile platform
and "αi" between the Z axis and the plane axes (X,
Y). The inverse geometric model can be expressed
by the following equations [21].
222 )()()( AizzAiyyAixxLi
;
i=1...5. (1)
)(arctan Aixx
Aiyy
gi
; i=1…5. (2)
)
)()(
(arctan 22 AiyyAixx
Aizz
gi
;
i=1…5. (3)
4 The Dynamic Modelling
In this section, we begin to present the dynamic
equation for a robot with five cables and its state-
space representation. Then, the response will be
simulated in a closed loop with a predictive
controller as a method [22].
4.1 Establishment of Mathematical Model
The dynamic model of the actuator can expressed
by the following equation according to the structure
pulley as illustrated in figure 2 [23]:
(4)
Where: m is the mass matrix and
X
is the
acceleration vector of the end-effector.
T
RzRyRxRFFFF
: is the resultant force
of all tensions applied to the cables.
Where:
Rz
Ry
Rx
F
F
F
z
y
x
m
m
m
00
00
00
(5)
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4.2 The Dynamic Comportment of the
Motors
Figure 3 shows the structure pulley and its dynamic
comportment is expressed by the following
equation:
.rTCJ
(6)
Fig. 3: Structure pulley.
with:
4
3
2
1
000
000
000
000
J
J
J
J
Jmat
and
4
3
2
1
000
000
000
000
C
C
C
C
Cmat
(7)
We consider that all the rays of the pulley are the
same:
ri = r(i=1.2...5),
T
i),....( 21
:is the vector of the torques applied
by the motors.
t(t1,t2,…ti)T :is the vector of tension cables.
β :is the angle of rotation of the pulley.
Θi :The angles between cables and the pulley.
So:
)(
1
CJ
r
t
. (8)
Where Li0 are the initial lengths of the cables:
222
0)()()( AizAiyAixLi
So
ii LL
LL LL
r
Xi
X
X
0
220
110
2
1
1
)(
)(
)(
. (9)
i=1,.. ,5
(10)
by subtracting successively (10) with respect to
time, we get:
x
x
x
xdt
d
(11)
Substituting (11) we obtain:
X
X
CX
X
X
Xdt
d
J
r
t



1
(12)
Finally, the set the equation of the dynamic
model can be expressed in a standard form for
robotic systems (13):
)()(),()()( 11 XSXMXXNXMtX
(13
)
Where:
X
JXSmrM

)(*
(14)
And
))((),(
X
X
C
Xdt
d
JXSXXN


(15)
i
2
1
i=1,…5 (16)
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Foued Inel, Mohammed Khadem,
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6 Control ONTROL Law and
Architecture
This section, the low-level control, has been
ensured by the development of a model Predictive
control based on the overall system Cartesian
Dynamics Equations of Motion, which is presented
in this section (Equation 13). In his paper on the
MPC gains are determined the error using a Matlab
program simulation to achieve reasonable
performance for the trajectories. The establishment
of the control law along X,Y and along Z is:
Consider the following MIMO nonlinear system:
Consider the following MIMO nonlinear system.
(17)
Where x Є Rn, U Є Rm and y Є Rm are,
respectively, the state vector, the control vector and
the output vector. F(x) and h(x) are smooth vector
fields.
Predictive controller is designed such that the
future output y (t+T) follows the future reference
signal yr (t+T) minimizing the quadratic
performance index given below:
(18)
Where:
e (t+T) is the future tracking error vector, and T >
0 is the prediction horizon.
The outputs of the system and the reference signals
predictions are approximated by their Taylor-series
expansions up to corresponding relative degree ρi:
(19)
(20)
The outputs yi time derivatives required in the
approximated performance index are ex-pressed as
given below using Lie derivatives [25]:
(21)
Introducing (21) in the approximated performance
index (22) yields:
(22)
Finally, the control law minimizing the
approximated performance index is obtained by
solving the system:
(23)
Consequently, the control law is given by [25]:
The control architecture as shown in Figure 4 is
made up of three different parts: the Predictive
controller, the tension calculation and pulley angle
β to determine the cable lengths Li.
Fig. 4: Control Architecture.
7 Simulation Results
In this part, we present the simulation of the
response for a 3D cables-based robot with 4 cables,
for dynamic equation, which has a non-linear
equation system, for this purpose, we use a Runge
Kutta method as a numeric solution. Runge and
Kutta developed the following formulae [26]:
y(x1) ≈y0+ (k1 + 2k2 + 2k3 + k4)/6,
k1 = hfo, , fo ≈ f(x0, y0),
k2 = hf (x0 + h/2, yo + k1/2),
k3= hf(x0 + h/2, y0 + k2/2),
k4 = hf(x0 + h, y0 + k3).
And we then implement a Cartesian Predictive
controller in this dynamic equation for reduce the
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Foued Inel, Mohammed Khadem,
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tracking error (er = desired – actual) for all the
axes. The parameters for the dynamics equations of
motion (15) for the 5cables are: point mass m =
0.01 kg; rotational shaft/pulley inertias Ji (i =
1,…,5) = 0.0008 kgm2; shaft rotational viscous
damping coefficients Ci (i = 1,.., 5) = 0.01 Nms
and ri = r = 1cm (for all i = 1,…, 5).
Therefore, we placed the system's reference in the
workspace's middle (0,0,0). Figures 5, 6 and 7
illustrate a graphical user interface for the point-to-
point command, allowing the user to input any
point's coordinates into the workspace, this last, the
end effector is moved precisely to this point when
the plot is clicked, and this interface can initialize
the robot's case (Figure 8). These techniques are
based on the inverse geometric model. Finally,
Figures 9 and 10 show the end effector's path for
different tests and plot the point when the values
for the z-axes are given.
Fig. 5: Plot the end effector's displacement to
position one
Fig. 6: Plot the end effector's displacement to
position two.
Fig. 7: Plot the end effector's displacement to
position three
Fig. 8: Plot initialization the end effector
When clicked on the plot, the end effector is
displaced directly to this target point with a high
precision, and also, this interface can initialize the
case of this robot (Figure 8), this technique based
on an inverse geometric model. Finally, Figures 9
and 10 show the end effector's path for different
tests and plot the point when the values for the z-
axes are given.
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Fig. 9: Plot the end effector's path for different
tests
Fig. 10: Plot the point when the values for the z-axe
is given
To illustrate the effectiveness of our control, we
use another interface graphic command that follows
a continuous trajectory. When we click on the plot,
the end effector moves and precisely follows the
predetermined trajectory in the workspace, as seen
in figure 11. Figure 12 shows the cable lengths
necessary to draw a spiral trajectory. According to
the results of various tests, this predictive control
performs better under most operating
conditions.
Fig.11: Plot the spiral trajectory
Fig. 12: Plot the cables lengths necessary to draw a
spiral trajectory
8 Conclusions
This paper presented a reel prototype of a novel 3D
cable base robot with five cables, this last, we have
applied simulation results for different tests. In this
way, we have designed predictive technique as a
control, then we developed an user interface
graphic with a simulation program to control the
displacement of end effector based on: point to
point command and according to the a predefined
trajectory, we assume that, the tensions values are
limited with tmin and tmax witch are always
positive and the cables lengths do not exceed the
workspace. Also, we have presented some results
for continuance trajectories. The simulation results
have demonstrated the effectiveness and feasibility
of the proposed control and are suitable for
improving the performance response.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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