Modeling of Nonlinear 3-RRR Planar Parallel Manipulator ...ijens.org/Vol_20_I_05/201505-4646...
Transcript of Modeling of Nonlinear 3-RRR Planar Parallel Manipulator ...ijens.org/Vol_20_I_05/201505-4646...
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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:05 175
201505-4646-IJMME-IJENS © October 2020 IJENS I J E N S
Modeling of Nonlinear 3-RRR Planar
Parallel Manipulator: Kinematics and
Dynamics Experimental Analysis
Abdelrahman Sayed Sayed 1,2 , Nada Ali Mohamed 1,2, Amr Abo Salem 1 and Hossam Hassan Ammar 1,2
1 School of Engineering and Applied Science, Nile University, 12588, Sheikh Zayed City, Giza, Egypt;
2 Smart Engineering Systems Research Center (SESC), Nile University, 12588, Sheikh Zayed City, Giza,
Egypt Emails : [email protected] (A.S.S), [email protected] (N.A.M), [email protected]
(A.A.S),
[email protected] (H.H.A)
Abstract— Parallel Manipulators (PMs) are gaining increasing importance, due to their superiority over serial manipulators in industry in terms of smaller workspace (WS), speed and precision. In this paper, the design, workspace analysis, modeling and control of a novel 3-RRR Planar Parallel Manipulator (PPM) are proposed. Because the kinematic constraint equations are complex due to the nonlinear behavior, non-conventional methods are used to model the system and control it. The Neuro- Fuzzy Inference System (NFIS) is used for forward kinematics modeling, while the Neural Networks (NN) and Adaptive Neuro- Fuzzy Inference System (ANFIS) are used for inverse kinematics modeling. Two metaheuristic optimization techniques, PSO and GA, were used for parameter tuning in the NFIS model. The results show that the used techniques were accurate in capturing the system dynamics. Thus, they enable precise and fast control using them, instead of using the coupled kinematical equations.
Index Term— Planar Parallel Manipulator (PPM), Kinematic and Dynamics analysis, Neural networks; ANFIS, Neuro-Fuzzy Inference System.
I. INTRODUCTION AND RELATED WORK Serial Manipulators are very important in industry, as they
have a lot of advantages such as large workspace, and ease of
control [1, 2]. Despite these advantages, there are
disadvantages such as small pay-load due to its serial
construction [3, 4]. Big errors as well at the end-effector due
to the error amplification in each joint. A serial manipulator is
referred as an open chain manipulator because it has one end
connected to ground and the other end is free to move.
Parallel Manipulators (PMs) are considered to have great
importance in many industrial and medical applications rather
than in flight simulations [5]. This importance came from its
design such as the length of the links, type of the joints,
position of the actuators, workspace, velocity and acceleration
of the end-effector [6]. PM can be defined as a closed chain
mechanism composed of an end-effector (EE) and a fixed base
linked together by at least two independent kinematic chains
[7]. The significance of parallel manipulator over serial one is
its higher payload-to-weight ratio [8], high accuracy, high
structural rigidity and low inertia of moving parts [9]. The
external loads can be divided and shared for more than one
actuator; PMs have a large capacity for carrying loads. On the
other hand, they suffer from relatively small workspaces,
complex kinematics and highly singular workspaces [10].
PPMs have high positioning accuracy because the joint errors
can be minimized. PPMs can replace some of the X-Y
machines such as the traditional CNC machine. Actuation
redundancy is one of the best technique to overcome some of
PPM’s issues [11]. Using this method, there are one or more
actuators added to the mechanism which can be used for
improving behavior through the whole workspace [12]. The
robotic mechanism could be chosen based on the task or the
type of work to be performed and determined by the position
of the robots and by their dimensions. In general, the selection
is done through experience, intuition and experiments of the
designer. It is necessary to formulate a quantitative measure of
the manipulation capability of the robotic system, which can
be useful in the robot control and in the trajectory planning.
The forward kinematic problem is described as finding the
actual pose of the end-effector in relation to the fixed base of
the robot from the active joints’ coordinates [13]. However,
computing the forward kinematic solution for parallel robots
are much more complicated than serial robots because of the
presence of a large number of independent joints’ equations
that describes the relation between the various links in the
robot [14]. Although closed-form solutions are preferred to be
used with PPM, the solution is usually complex and in some
cases it does not exist [15]. Another type of solutions was
based on numerical methods such as Newton-Raphson method
and learning networks. But, these methods were computation-
ally intensive when implemented and it does not guarantee to
compute all of the possible solutions [16]. A solution to such
problems is to implement a machine learning technique, as
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these techniques have the ability to analyze many massive pre-
defined data sets in a process similar to the learning process of
the human. So, combining such techniques with the
mathematically complex parallel planar robot would help to
save time in setting up and cloning such robots in many fields.
Neural Networks are a set of algorithms that were developed
from mimicking the actual human neural network in terms of
predicting and detecting similar patterns. An adaptive
network- based fuzzy inference system (ANFIS) can be used
with specified membership functions to estimate the inverse
kinematic behavior of the parallel planar manipulator [17].
Other revolutionary metaheuristic optimization techniques
were used to train neuro fuzzy inference system (ANFIS)
using data from the actual proposed design.
This paper is organized as follow. In section 2, the
mechanism of the 3-RRR Planar Parallel Manipulator (PPM)
is introduced, in section 3, forward and inverse kinematics
have been deduced. Then, in section 4, the dynamics of the
redundant 3-RRR Planar Parallel Manipulator (PPM) is
discussed. Non- conventional modelling methods including
NN, ANFIS and NFIS are presented in Section 5.
Experimental results and discussions are described in Section
6. Finally, conclusions are presented in Section 7.
II. PLANAR PARALLEL MANIPULATOR’S MECHANISM
Researchers have focused their works for spatial parallel
manipulators but usage of such planar configurations may not
be neglected. To understand the parallel manipulators, we
should go through the mechanic motion of the closed chain
mechanism, it is found that the 4-bar mechanism with only one
actuator produces a definite motion with a definite trajectory.
Joints could be one of two types, the revolute one which is a
hinged joint, referred by (R) or a prismatic one which is a
sliding joint, denoted by (P). The link that connected to ground
by a revolute joint is called a crank. A link connected to
ground by a prismatic joint is called a slider. In some times,
sliders are considered to be cranks that have a hinged pivot at
an extremely long distance away perpendicular to the travel
of the slider. These configurations produce such 3-RRR. This
manipulator has a planar motion with planar workspace.
The 3-RRR PPM consists of the 3 arms as shown in Figure
1. Each arm is consisted of one active joint 𝐴𝑖 where 𝑖 =1,2,3 for arms 1, 2, 3 respectively. One passive joint 𝐵𝑖, one active link 𝑟𝑖 - which connects the active joint Ai to the passive joint 𝐵𝑖. One passive link 𝑙𝑖 - which connects the passive joint 𝐵𝑖 to the EE. The three active joints 𝐴𝑖 are located at the apex of an equilateral triangle, and the end of the three arms are
connected together at a point to produce the EE of our
mechanism. All angles are referred to the x-axis and CCW for
positive direction.
There are a lot of applications such 3-RRR planar parallel
configurations are pick and place operation over a plane
surface, machining of plane surfaces and mobile base for a
spatial manipulator. The majority of studies on PM, unlike
serial manipulators, however, the redundancy can be divided
into actuation redundancy and kinematic redundancy.
Actuation redundancy can be explained as replacing existing
passive joints of a manipulator by active ones. Redundant
actuation has been recently proposed as an efficient method
to improve the performances such as workspace, stiffness and
dexterity of parallel manipulators. Consequently, actuation
redundancy does not change the mobility or force workspace
of a manipulator but may cause singularity reduction.
Kinematic redundancy take place when an extra active joints
and links are inserted to manipulators. Adding kinematic
redundancy in parallel manipulators have many advantages
such as avoiding most kinematic singularities, enlarging the
workspace, and also improving dexterity [11, 18]. However,
when a manipulator is kinematically redundant, the
complexity of actuation schemes is greater since there is an
infinite number of solutions for the inverse displacement
problem, and therefore, an infinite choice of actuation
schemes. To investigate the 3-RRR, it is essential to analyze
the inverse kinematics and the workspace. PMs can give both
multiple inverse kinematic solutions and multiple direct
kinematic solutions.
Fig. 2. The 3-RRR PPM Cutting Out
Fig. 1. The 3-RRR PPM
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201505-4646-IJMME-IJENS © October 2020 IJENS I J E N S
III. KINEMATIC MODEL OF 3-RRR PPM
A. Forward Kinematic Model (FKM)
As shown in Figure 1, using the mathematics of
trigonometrical laws for the triangles A1B1C, A2B2C
and A3B3C, it is found that:
(𝐶𝑥 − 𝑏1𝑥)2 + (𝐶𝑦 − 𝑏1𝑦)
2= 𝑙1
2 (1)
(𝐶𝑥 − 𝑏2𝑥)2 + (𝐶𝑦 − 𝑏2𝑦)
2= 𝑙2
2 (2)
(𝐶𝑥 − 𝑏3𝑥)2 + (𝐶𝑦 − 𝑏3𝑦)
2= 𝑙3
2 (3)
By solving the previous equations eq. 1, 2 and 3, one can get
the following:
𝐺1𝐶𝑥 + 𝐺2𝐶𝑦 + 𝐺3 = 𝑙22 − 𝑙1
2 (4)
𝐺4𝐶𝑥 + 𝐺5𝐶𝑦 + 𝐺6 = 𝑙32 − 𝑙1
2 (5)
After putting these previous equations eq. 4 and eq. 5 in matrix
form, one can get that:
[𝐶𝑥𝐶𝑦
] = [𝐺1 𝐺2𝐺4 𝐺5
]−1
[𝑙22 − 𝑙1
2 − 𝐺3𝑙32 − 𝑙1
2 − 𝐺6] (6)
where:
𝐺1 = 2𝑏1𝑥 − 2𝑏2𝑥
𝐺2 = 2𝑏1𝑦 − 2𝑏2𝑦
𝐺3 = (𝑏2𝑥2 + 𝑏2𝑦
2 ) − (𝑏1𝑥2 + 𝑏1𝑦
2 )
𝐺4 = 2𝑏1𝑥 − 2𝑏3𝑥
𝐺5 = 2𝑏1𝑦 − 2𝑏3𝑦
𝐺6 = (𝑏3𝑥2 + 𝑏3𝑦
2 ) − (𝑏1𝑥2 + 𝑏1𝑦
2 )
Notice that [𝑏𝑖𝑥 𝑏𝑖𝑦]𝑇
= [𝑎𝑖𝑥 + 𝑟𝑖𝐶𝛼𝑖 𝑎𝑖𝑦 + 𝑟𝑖𝑆𝛼𝑖]𝑇 can be
solved when the actuation at the first joint of each link, where: 𝐶𝛼𝑖 = cos(𝛼𝑖) and 𝑆𝛼𝑖 = sin (𝛼𝑖).
B. Inverse Kinematic Model (IKM)
Using the Law of Cosines in triangle A1B1C shown in Figure 1, there will be a relation between the angle 𝛼𝑖, the angle between the x-axis and the active link 𝑟𝑖 and the EE point C where it can be found as:
(𝐵1𝐶)2 = (𝐴1𝐵1)
2 + (𝐴1𝐶)2
− 2(𝐴1𝐵1)(𝐴1𝐶)cos(𝛼1 − 𝜃1)
∴ 𝛼1 = cos−1 (
(𝐴1𝐵1)2 + (𝐴1𝐶)
2 − (𝐵1𝐶)2
2(𝐴1𝐵1)(𝐴1𝐶)) + 𝜃1
(7)
𝜃1 = tan−1 (
𝐶𝑦 − 𝑎1𝑦𝐶𝑥 − 𝑎1𝑥
) (8)
where the 𝜃1 is the angle between the x-axis and dashed line between point A1 and the EE. Notice that, 𝐴1𝐵1 = 𝑟1, 𝐵1𝐶 = 𝑙1 and it is easy to determine A1C and it will be as follow:
𝐴1𝐶 = √(𝐶𝑦 − 𝑎1𝑦)2+ (𝐶𝑥 − 𝑎1𝑥)
2 (9)
The similar derivation can also be used for determining 𝛼2 and 𝛼3 as follow:
𝛼2 = cos−1 (
(𝐴2𝐵2)2 + (𝐴2𝐶)
2 − (𝐵2𝐶)2
2(𝐴2𝐵2)(𝐴2𝐶)) + 𝜃2 (10)
𝛼3 = cos−1 (
(𝐴3𝐵3)2 + (𝐴3𝐶)
2 − (𝐵3𝐶)2
2(𝐴3𝐵3)(𝐴3𝐶)) + 𝜃3 (11)
Now, it is easily to simulate the 3-RRR Using these
previous equations and MATLAB software as shown in
Figure 3. Also, the boundaries of workspace can be deduced
for the 3-RRR as shown in Figure 4. This area of the WS can
be done by extending the three arms and draw three circles
intersected which produce the WS. Note that, the red dot
circles in Figure 4 refer to the active joints 𝐴𝑖.
Fig. 3. The 3-RRR PPM using MATLAB
Fig. 4. WS for the 3-RRR PPM
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201505-4646-IJMME-IJENS © October 2020 IJENS I J E N S
IV. DYNAMICS OF REDUNDANT PARALLEL MANIPULATOR To derive the dynamic model of the redundant actuation 3-
RRR PPM, the Lagrange D’Alembert formulation is used. It
is assumed a virtual cut at the point of the EE to eliminate the
mechanical constraints for the 3-RRR PPM as shown in Figure
2. The next step is by applying the Lagrange equation for the
open loop or chain mechanism. Then, adding the constraint
matrix in specific form to the model in order to get the 3-RRR
PPM. For that, the following illustrates these steps.
A. Dynamic Modeling
One can derive the dynamic model for the open chain
mechanism starting from the kinetic and potential energies
[18]. Euler Lagrange formulation permits modeling for each
branch of the open chain mechanism as shown in Figure 2. In
the modeling, consider the horizontal motion of the PM. The
following equations, illustrate the derivation of the model for
each arm.
𝑀𝑖 [�̈�𝑖𝛽𝑖
] + 𝐶𝑖 [�̇�𝑖�̇�𝑖
] = [𝜏𝛼𝑖𝜏𝛼𝑖
] (12)
where:
𝑀𝑖 = [𝑚𝑖1 𝑚𝑖2𝑚𝑖3 𝑚𝑖4
] = [𝜆𝑖 𝜎𝑖 cos(𝛼𝑖 − 𝛽𝑖)
𝜎𝑖 cos(𝛼𝑖 − 𝛽𝑖) 𝛾𝑖]
𝐶𝑖 = [𝑐𝑖1 𝑐𝑖2𝑐𝑖3 𝑐𝑖4
] = [0 𝜎𝑖 sin(𝛼𝑖 − 𝛽𝑖) �̇�𝑖
𝜎𝑖 sin(𝛼𝑖 − 𝛽𝑖) �̇�𝑖 0]
𝜆𝑖 = 𝐽𝑖1 + 𝑚𝑖1𝑟𝑖12 + 𝑚𝑖2𝑙
2
𝛾𝑖 = 𝐽𝑖2 + 𝑚𝑖2𝑟𝑖22
𝜎𝑖 = 𝑚𝑖2𝑙𝑟𝑖2
where: lengths of the links for the manipulator are equal as
shown in (𝑙 = 𝑟𝑖 = 𝑙𝑖), the variables 𝑚𝑖1 and 𝑚𝑖2 refer to the masses of the links, the variables 𝐽𝑖1 and 𝐽𝑖2 refer to the moments of inertia of the links relative to the mass centers, the
variables 𝑟𝑖1 and 𝑟𝑖2 refer to the distances between mass centers and joints of links, and l refers to the length of links.
Now, by merging those equations for the open chain
mechanism, one can get the following:
𝑀(𝑞)�̈� + 𝐶(𝑞, �̇�)�̇� = 𝜏 (13)
where 𝑀(𝑞) is the inertial matrix, 𝐶(𝑞, �̇�) is the Coriolis matrix. The following mathematical expressions describe the
symbols in Eqn. 13.
𝑞 = [𝛼1 𝛼2 𝛼3 𝛽1 𝛽2 𝛽3]𝑇
𝜏 = [𝜏𝛼1 𝜏𝛼2 𝜏𝛼3 𝜏𝛽1 𝜏𝛽2 𝜏𝛽3]𝑇
Then, Eqn. 14 and Eqn. 15 shows the expressions for matrix
M and matrix C, where 𝑑𝑖 = 𝛼𝑖 − 𝛽𝑖 .
𝑀 =
[
𝜆1 0 00 𝜆2 00 0 𝜆3
𝜎1 cos(𝑑𝑖) 0 00 𝜎2 cos(𝑑2) 00 0 𝜎3 cos(𝑑3)
𝜎1 cos(𝑑𝑖) 0 00 𝜎2 cos(𝑑2) 00 0 𝜎3 cos(𝑑3)𝛾1 0 00 𝛾2 00 0 𝛾3 ]
(14)
𝐶 =
[
0 0 00 0 00 0 0
𝜎1 sin(𝑑𝑖) �̇�1 0 0
0 𝜎2 sin(𝑑2) �̇�2 0
0 0 𝜎3 sin(𝑑3) �̇�3
𝜎1 sin(𝑑𝑖) �̇�1 0 0
0 𝜎2 sin(𝑑2) �̇�2 0
0 0 𝜎3 sin(𝑑3) �̇�30 0 00 0 00 0 0 ]
(15)
B. Redundant Actuating Modeling
Proposition 1: Assume that the joint torque of the
equivalent open-chain system for a given motion which
satisfies the loop constraint is given by τ. Then, the joint torque
of the redundantly actuated closed-chain system required to
generate the same motion is given by Eqn. 16.
𝑊𝑇𝜏 = 𝐽𝑇𝜏𝛼 (16)
where 𝑊 = 𝜕𝑞 𝜕𝑥⁄ and 𝐽 = 𝜕𝑞𝛼 𝜕𝑥⁄ . In Figure 2, it shows the equivalent open-chain system for the manipulator 3-RRR
PPM. Let the generalized coordinates be q, the 𝑞𝛼 is the vector of the active joints and 𝑞𝛽 is the vector of the passive joints.
Also, if τ the generalized torque, let the active joint’s torque
will be 𝜏𝛼 and the passive joint’s torque will be 𝜏𝛽. For
actuation redundancy situation, Eqn. 13 is not correct and to
prove that, the following constraints in Eqn. 17 should be
considered.
𝑞𝛼 = 𝑞𝛼(𝑥), 𝑞𝛽 = 𝑞𝛽(𝑥) (17)
Then, by applying differentiation on Eqn. 17, the Eqn. 18 results.
𝛿𝑞𝛼 =𝜕𝑞𝛼𝜕𝑥
𝛿𝑥, 𝛿𝑞𝛽 =𝜕𝑞𝛽
𝜕𝑥𝛿𝑥
(18)
Using Eqn. 18 and applying D’Alembert principle as shown
in Eqn. 12, one can deduce the relation shown in Eqn. 19.
[(𝑑
𝑑𝑡(
𝜕𝐿
𝜕�̇�𝛼) −
𝜕𝐿
𝜕𝑞𝛼− 𝜏𝛼
𝑇)𝜕𝑞𝛼𝜕𝑥
+
(19)
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(𝑑
𝑑𝑡(
𝜕𝐿
𝜕�̇�𝛽) −
𝜕𝐿
𝜕𝑞𝛽− 𝜏𝛽
𝑇)𝜕𝑞𝛽
𝜕𝑥] 𝛿𝑞 = 0
Because 𝛿𝑥 is free to vary, so Eqn. 20 results.
[𝑑
𝑑𝑡(
𝜕𝐿
𝜕�̇�𝛼) −
𝜕𝐿
𝜕𝑞𝛼,𝑑
𝑑𝑡(
𝜕𝐿
𝜕�̇�𝛽) −
𝜕𝐿
𝜕𝑞𝛽 ] [
𝜕𝑞𝛼𝜕𝑥𝜕𝑞𝛽
𝜕𝑥
]
= (𝑑
𝑑𝑡(𝜕𝐿
𝜕�̇�) −
𝜕𝐿
𝜕𝑞)𝜕𝐿
𝜕𝑥= 𝜏𝛼
𝑇𝜕𝐿
𝜕𝑥+ 𝜏𝛽
𝑇𝜕𝐿
𝜕𝑥
(20)
As 𝜏𝛽 ignored and assumed to be zero, so:
𝑊𝑇𝜏 = 𝐽𝑇𝜏𝛼 (21)
The dynamic model of each branch as an open chain, can be formed as in Eqn. 22.
𝑀(𝑞)�̈� + 𝐶(𝑞, �̇�)�̇� = 𝜏 (22)
By using Eqn. 22 and substituting in Eqn. 21. It is found that:
𝑊𝑇(𝑀(𝑞)�̈� + 𝐶(𝑞, �̇�)�̇�) = 𝐽𝑇𝜏𝛼 (23)
By differentiating the kinematics constraints, Eqn. 24 results.
�̈� = �̇��̇� + 𝑊�̈� (24)
By substituting with Eqn. 24 in Eqn. 23, the result is expressed
in Eqn. 25.
(𝑊𝑇𝑀(𝑞)𝑊)�̈� + (𝑊𝑇𝑀(𝑞)�̇� + 𝑊𝑇𝐶(𝑞, �̇�)𝑊)�̇�= 𝐽𝑇𝜏𝛼
(25)
�̂��̈� + �̂��̇� = 𝐽𝑇𝜏𝛼 (26)
Matrices �̂� and �̂� have the following properties as long as matrix W is full rank:
1. M is a symmetric and positive definite matrix. 2. M − 2C is a skew-symmetric matrix.
To get the constraint matrix W, differentiation of the
constraint equations for the whole system is done to get:
𝑊 =
[
𝜇1 cos(𝛽1) 𝜇1 sin(𝛽1)𝜇2 cos(𝛽2) 𝜇2 sin(𝛽2)𝜇3 cos(𝛽3) 𝜇3 sin(𝛽3)
−𝜇1 cos(𝛼1) −𝜇1 sin(𝛼1)−𝜇2 cos(𝛼2) −𝜇2 sin(𝛼2)−𝜇3 cos(𝛼3) −𝜇3 sin(𝛼3)]
where:
𝜇𝑖 =1
𝑙 sin(𝛽𝑖 − 𝛼𝑖)
C. Dynamic Performance Analysis
No one can deny that, there is a difference between the
simulation and practical work results especially when the talk
is about the PMs’. Parallel manipulators have a feature of
perfect dynamics with high degree. This feature cannot be
represented because the dynamic characteristics could not be
involved in the design. It is only considered in the model phase
for the controller. It will be better if the dynamic
characteristics merged in the step before the implementing the
PM. For that, analyzing the dynamic motion for the EE
making it easily to accelerate in the direction of the major axis
of the inertia ellipsoid. On the other hand, it is difficult to
accelerate in the direction of minor axis. The lengths of the
principal axes of the inertia ellipsoid can be referred as the
maximum and minimum singular values of inertia matrix. The
performance of the acceleration is isotropic if and only if, the
lengths of the axes are the same. If there are any differences in
the lengths, then, there is anisotropic for the accelerating
performance.
The accelerating/decelerating capabilities along all
directions should be more isotropic, consequently, the
condition number of the inertia matrix in the dynamic
equation, i.e., 𝐾𝐷, is proposed to quantify the dynamic manipulability of manipulators. The definition of 𝐾𝐷 is shown in Eqn. 27.
−1 ≤ 𝐾𝐷 =𝜎𝑚𝑖𝑛𝜎𝑚𝑎𝑥
≤ 1 (27)
where 𝜎𝑚𝑖𝑛 and 𝜎𝑚𝑎𝑥 are the minimum and maximum singular values of the inertia matrix, respectively. In order to
investigate the performance of the designed manipulator, the
condition number of Jacobian matrix and the distribution of
𝐾𝐷 of Eqn. 27 are given, as shown in Figure 5. One may see that the condition number distribution and the distribution of
𝐾𝐷 are symmetrical and the mechanism has a desired condition performance. From the previous Figure 5, it is found
that there is a symmetry at the actuators area, and also the zero
line is in the center of the workspace. It means that the 3-RRR
PPM has a better dynamic index in the center.
Fig. 5. Dynamic Index for the 3-RRR PPM
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V. NON CONVENTIONAL KINEMATICS MODELING METHODS
The need to use unconventional modeling methods arises
from the need to replace the derivation of direct and inverse
kinematics for each robotic model. Because analytical
derivation depends on the robot dimensions and structure, it
can be a tedious task in some cases [19]. Thus, the following
proposed methods are used to train an algorithm to learn the
kinematics from a set of observations. Nevertheless, this can
be applied to forward and inverse kinematics as shown below.
A. Neural Networks structure and parameters (NN)
Neural Networks is a large class of algorithms that mimics
the idea of brain in living organisms. The nodes in the neural
networks represent the neurons while the data trans- mitted
between them represent the neural signal. The use of multiple
neurons as well as different layers of neurons are all done to
better process the data and decrease the error. The
computational time for such unconventional method of
modeling is always questionable; however, given the recently
developed algorithms and the current PCs specifications, the
running time for such technique becomes reasonable, with
respect to the accuracy it provides. In our simulation, 3 input
layers, 3 hidden layers and 3 output layers where used. The
hidden layers contain a total of 35 neurons. MATLAB Parallel
Computing Toolbox and MATLAB Neural Networks Toolbox
were used in the simulation, using the Levenberg- Marquardt
algorithm. The mean squared error (MSE) was used for
measurement of results performance measurement, and the
stopping criteria was based on reaching a MSE value of less
than 1E-20. Because the inverse kinematics is the analysis
subject, the inputs are the x, y and theta of the end effector,
while the outputs are the three joint angles satisfying the end
effector position and orientation.
B. Adaptive Neuro-Fuzzy Inference System structure and parameters (ANFIS)
The Adaptive Neuro-Fuzzy Inference System is based on
the NFIS structure, and it is similar to it in combining
properties and performances of neural systems and fuzzy
logic. The modification of this structure to be adaptive
accounts to the higher precision in results that can be obtained
from using it as a replacement for modeling. Moreover, this
algorithm was subjected to a series of modifications as shown
in [20, 21]. Similar to NFIS, three different models for ANFIS
were developed to account for the three outputs. How- ever,
here the inputs are the pose of the triangular end effector and
the outputs are the three joints angles. Throughout our
simulation, 5 Gaussian membership functions for each input
and linear output functions are used for the fuzzy inference.
Moreover, 125 firing rules were used and they are allowed to
change during running to reduce the final error. Table 1 shows
the parameters used to train ANFIS. The stopping criteria was
based on the earlier one of two events: reaching the maximum
epoch of 10 or reaches an error of less than 1E-20. The relation
between the input data and the output data in our model is
shown in the model structure in Figure 6.
C. Neuro-Fuzzy Inference System structure and Metaheuristic optimization algorithms (NFIS)
The NFIS algorithm takes the advantage of both neural
networks and the fuzzy logic. Neural networks are
characterized by having multiple layers between which the
input data are processed and transferred until forming the
outputs layer. On the other hand, the fuzzy logic is
characterized by categorizing the inputs according to some
membership functions, and giving one output per model.
Table I
ANFIS Parameters
ANFIS Parameter Type Value
TSK Type Zero-order
Number of rules 125
Training epochs 2000
Nodes 286
Total fitting parameters 530
Premise (nonlinear) parameters 30
Consequent (linear) parameters 500
Membership function 5
Defuzzification method Weighted-average
Initial step size, 𝑲𝒊𝒏𝒊 0.01
Step increasing rate, 𝜼 1.1
Step decreasing rate, 𝜸 0.9
The output of fuzzy logic lies between 0 and 1. In such
setting, three models were constructed for our 3-RRR robot,
where the inputs of all the models are the joint angles, and the
outputs represented in x, y and theta of the end effector are
divided upon the three models. In order to tune the NFIS
model, two metaheuristic techniques were used, they are
Particle Swarm Optimization (PSO) and Genetic Optimization
Algorithm (GA). Table II shows the parameters used to train
the NFIS model with PSO and GA algorithms.
a) Particle Swarm Optimization (PSO): This is an iterative optimization approach that aims to optimize variables
in a problem, which are the tuning parameters for the NFIS
system here. As the name suggests, this method mimics the communication pattern between swarms of birds, in which the
Fig. 6. ANFIS Model Structure
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state and environmental perception of each member can be
shared with all other members to benefit the swarm as a whole,
especially if they collaboratively work towards a single task.
As Eqn. 28 shows, the new position of each particle 𝑖, 𝑥𝑖(𝑡 +1), depends on the past position, 𝑥𝑖(𝑡), and the velocity, 𝑣𝑖(𝑡).
𝑥𝑖(𝑡 + 1) = 𝑥𝑖(𝑡) + 𝑣𝑖(𝑡 + 1) (28)
Table II
NFIS Tuned with Metaheuristic Techniques
Parameter Value
Fuzzy structure Sugeno-type
Types of MFs Gaussian, Triangular
Output MF Linear
No. of inputs 3
No. of outputs 1
No. of fuzzy rules 24
Epochs number 1000
No. of samples 600
No. of population 50
b) Genetic Optimization Algorithm (GA): This is an iterative optimization method based on the evolution idea, and
more specifically on natural selection shown in Darwin’s work.
As apparent in the natural selection of organisms, this
optimization algorithm tries to get better solutions, called
generations, according to characteristic variables, called genes.
A further division of genes into a unique string that distinguish
each solution is done. These strings are called chromosomes,
and Eqn. 29, where N(s, t + 1) is the number of chromosomes in the next generation, N(s, t) is the number of chromosomes in the current generation, 𝑢(𝑠, 𝑡) is the average fitness of the chromosomes and s is the schema. The algorithm starts with
an initial set of solutions that solve the problem to different
extents, then the algorithm tries to compare the fitness values
of these solutions to optimize the problem. Logically, the
individuals with better genes, represented in better fitness
score, are selected to pass their genes and chromosomes to
the next generation. Mimicking the natural selection, crossover
takes place to get fitter individuals in the next generation.
Then, the process is iterated on the next generation. In our
simulation, the maximum number of generations was set to
1000. However, the simulation can stop before that if the fitness value reaches a certain prescribed limit.
𝑁(𝑠, 𝑡 + 1) = 𝑢(𝑠, 𝑡)[1 − 𝑒]𝑁(𝑠, 𝑡) (29)
VI. EXPERIMENTAL RESULTS AND DISCUSSION
The experimental platform is a 2-DOF parallel manipulator
with three actuators together with three internal encoders as
shown in Figure 7. As a result, the three 2-link robots,
constrained in such a way that their end-effectors (the multiple
joints) coincided with one another. The 2-DOF redundantly
actuated PPM has been built. It consists of base with area
1 𝑚2. Three EC MAXON motors with 250 watts have been placed at the vertex of equilateral triangle with side length
70 𝑚𝑚. These three motors are implemented for the active joints with reduction ratio 74:1. Each motor has its own
EPOS 70/10 driver. Each motor connected to active link
with length 40 𝑚𝑚 and each active link connected to passive link with length 40 𝑚𝑚. This combination produces three arms, which connected together at the end-effector point
to produce the redundant actuated 3-RRR PPM. For EPOS
70/10 drivers, they connected to PC using three serial port
communication. MATLAB, LABVIEW and EPOS MAXON
library have been installed on PC with Microsoft Windows XP
Professional Version 2002 Service Pack 3, Pentium processor
[email protected] and 4GB of RAM. In Figure 8, one can find
the system hardware implementation as explained.
The data used was obtained from the Motion Analysis of
the experimental setup and using color object tracking based
on computer vision to detect the end-effector position and also
collect the motors encoder angles. Starting with the results for
the forward kinematics, the NFIS model was based on 3
membership functions as shown in Figure 9 for the 3 inputs of
the system. Each output can be represented by a rule base
surface to visualize the optimization dynamics. As expected,
the PSO and GA tunings were applied on the same dataset for
consistence and reliability; this dataset contains 600 sample
after filtration. The results of the simulation were shown in
Figures 10 to 11, where the actual data was plotted for
comparison between the PSO algorithm and the GA algorithm
to know which one fits the data better. The three outputs
represented the three NFIS models used. By analyzing the
Fig. 7. 3-RRR Planar Parallel Manipulator
Fig. 8. Hardware Implementation of 3-RRR Planar Parallel Manipulator
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results, the GA algorithm is apparently better in performance
by having less mean error, greater stability and low
computational time.
By focusing on the error of training, Tables III and IV
shows the performance measurements for the PSO and GA
optimizations with the NFIS model, respectively. These
performance measurements include mean absolute error, root
mean square error, Pearson Rp, etc. Accurately speaking, the
figures with the tables reveal that both models succeeds in
capturing the kinematical behavior with slight error between
them, making the decision of which is better hard. For
example, the GA optimization was better in predicting the end
effector angle; however, PSO optimization was better in
predicting the position of the end effector. Also, what is better
depends on the performance measurement used.
Moving to the inverse kinematics and the two used
techniques, namely NN and ANFIS. First, both models were
able to learn and validate the data from two data sets for
consistency. The first contains 1142 samples after filtration
and it was used in training and testing, while the second
contains 1039 samples and it was used in validation.
Table III
statistical analysis for NFIS tuned with PSO model
During the run time, it was clear that NN model outperform
the ANFIS model greatly in terms of the computational time.
The former took around 250 seconds to learn and give
prediction, while the latter took around 20500 seconds for the
same task. This is the first advantage for the NN algorithm.
Then, the results of both models were visualized while Figures
12 to 14 show the results of NN algorithm on the training and
testing set, while Figures 15 to 17 show the results of ANFIS
algorithm on the same set. Then, by applying the same
predicted model on a new data set called the validation set, the
statistics shown in Tables V and VI are summarizing the
performance measurement of NN and ANFIS model,
respectively, according to different measures such as, root
mean square error, Pearson Rp, etc. These tables are also
comparing between the behaviors of the same model under the
training verses the validation data set.
X Y
Mean 0.2347 -0.0175
SD 2.5253 1.5053
MAE 1.6773 1.0438
RMSE 2.5437 1.5102
NRMSE 0.0096 0.0063
Pearson 𝑹𝒑 0.9986 0.9996
Pearson 𝑹𝟐𝒑 0.9976 0.9993
S-Rs 0.9963 0.9911 8
Fig. 9. Input membership functions (MFS)
Fig. 10. NFIS X-axis output results
Fig. 11. NFIS X-axis output results
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Table IV
statistical analysis for NFIS tuned with GA model
Table V
Statistical analysis for NN model
θ1 θ2 θ3
Testing Validation Testing Validation Testing Validation
Mean ± SD 182.85 ± 147.18 182.42 ± 147.05 124.5 ± 42.6 132.85 ± 48.95 274.15 ± 75.2 269.05 ± 77.72
MAE 24.7 88.9 15.4 31.8 17.5 33.2
RMSE 47.9 137.7 20.8 43.5 29.7 53.5
NRMSE 0.26 0.96 0.17 0.37 0.11 0.19
Pearson Rp 0.95 0.58 0.89 0.59 0.93 0.78
Pearson 𝑹𝟐𝒑 0.89 0.33 0.81 0.35 0.86 0.60
S-Rs 0.84 0.54 0.91 0.63 0.83 0.56
Table VI
Statistical analysis for ANFIS model
θ1 θ2 θ3
Testing Validation Testing Validation Testing Validation
Mean ± SD 184.33 ± 149.3 138.4 ± 1184.5 124.1 ± 47.3 141.4 ± 283.2 275.8 ± 79.2 311.74 ± 822.5
MAE 11.7 356.8 2.3 84.9 4.94 176.1
RMSE 21.26 1169.5 4.44 283.2 8.63 815.6
NRMSE 0.12 8.16 0.04 2.41 0.03 3.03
Pearson Rp 0.99 0.16 0.99 0.09 0.99 0.14
Pearson 𝑹𝟐𝒑 0.98 0.03 0.99 0.01 0.98 0.02
S-Rs 0.92 0.47 0.99 0.48 0.97 0.39
Fig. 12. NN output 1 results vs the actual output
Fig. 13. NN output 2 results vs the actual output
X Y
Mean 0.0086 -0.0578
SD 1.9939 2.9579
MAE 1.3246 1.6760
RMSE 1.8932 2.9561
NRMSE 0.0075 0.0121
Pearson 𝑹𝒑 0.9953 0.9974
Pearson 𝑹𝟐𝒑 0.9985 0.9978 S-Rs 0.9975 0.9946
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Fig. 14. NN output 3 results vs the actual output
Fig. 15: ANFIS output 1 vs the actual output
Fig. 16. ANFIS output 1 vs the actual output
Fig. 17. ANFIS output 1 vs the actual output
These results point out clearly the second advantage of the
NN over the ANFIS model in being able to better fit the model
using the two sets in our analysis. This result was apparent in
the validation data set more than in the training one. For
example, in predicting the first joint angle, the NN model has
a Pearson correlation of 0.575, while the ANFIS model has the
correlation of 0.1597. Accordingly, it was proved that the
difference between NN and ANFIS model, having same
running parameters and data sets, differ greatly from each
other, in favor of the NN model which performs better in the
problem of inverse kinematics.
VII. CONCLUSION
Robotics manipulators are of a great use in many different
fields and applications. While the serial manipulator is widely
used in industry with many well-known models, the parallel
manipulator is extremely flexible in achieving fine tuning of
end-effector angle. However, the modeling of parallel
manipulator is more complicated than the serial one. In this
paper, neural networks, with 4 different algorithms, were used
to replace the modeling of the parallel manipulator and helped
in solving forward and inverse kinematics problems. For
forward kinematics, NFIS structure algorithm was used and
validated. For inverse kinematics, NN structure and ANFIS
were used and validated. In both cases, and both models in
each case, the unconventional methods for modeling succeed
in capturing the kinematics behavior of the robotic
manipulator. The limitation of this study lies in the
experimental testing of the methods using the robot itself
along with a camera to detect the end- effector position and
orientation accurately. The future work in this area is
represented in analyzing the dynamics of the systems and
modelling it in order to have a bigger idea about the actual
control using motor inputs and response of the system.
VIII. ACKNOWLEDGMENT
The authors would like to thank Renewable Energy and
Modern Control Lab, Nile University for facilitating all the
procedures required to complete this study.
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