Modeling of Nonlinear 3-RRR Planar Parallel Manipulator ...ijens.org/Vol_20_I_05/201505-4646...

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



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



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)



(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



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



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



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



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


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




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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