PSO-RBNN Based Control Design for Trajectory Tracking

NEHA KHURANA

Maharishi Dayanand University, Haryana, INDIA.

Abstract: Inspite of so universally accepted, control performance by NN depends on many of the varying factors such

as output weights. To ensure the functional accuracy of the NN, it is required to have an defined value of these

performance effecting factors. Control scheme proposed in this paper uses an emerging optimization technique

naming, PSO to get the optimal value of the parameters, naming spread factor and weights of output layer in RBNN.

Thus, this hybrid controller possesses the advantageous qualities of RBNN and PSO both. For the further improvement

in the basic PSO algorithm, inertia weight factor of PSO is made adaptive.This projected controller has been verified

by comparing it with a basic PSO and the basic RBNN controller for the trajectory tracking control of a 2-DOF

remotely driven robotic manipulator. To check the robustness of the controller its performance has been checked by

incorporating uncertainties naming payload masses and friction. Appropriate conclusions have been drawn in last.

Keywords: Radial Bias Neural Network (RBNN), Particle Swarm Optimization (PSO), Evolutionary Neural Network

(ENN), Hybrid Intelligent Controller, Remotely Driven Links Manipulator, Motion Control of Non-linear systems

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1. Introduction

With the increase in present applications of computers and its

everyday increasing future prospects; areas of artificial

intelligence based controllers have been expanded exponentially.

Since the last few decades, because of highly non-linear mapping

capabilities, neural network is one of the most widely used AI

techniques [1]. There are a wide number of types of neural

networks proposed in literature. Each one has its own advantages

and disadvantages. In terms of time-taken, accuracy in results and

non-linear mapping capabilities for non linear systems like

motion control of robotic manipulator, RBNN (Radial Bias

Neural Network) is found to superior when compared with back

propagation neural network [2-8]. RBNN is given by Broomhead

and Lowe [9], and its interpolation and generalization properties

are thoroughly investigated in [10, 11]. As stated, although

RBNN is one of the commonly used NN based control scheme for

the non linear, time varying control system, yet the accuracy in

performance of RBNN depends mainly upon the specific values

of some of its parameters. A few of the important performance

deciding RBNN factors are spread factor, (𝜎𝑗 ) and weights from

hidden to output layer, (wjk). Most favorable value of these

parameters can be chosen by either some expert’s experience or

by trial and error (TAE) method. This limitation of RBNN

restricts its use to an expert or by using time consuming, tedious

and frustrating TAE method by an amateur. This limitation of

RBNN restricts its use or deteriorates its performance. From the

above discussions it can also be inferred that improvements in

RBNN can be made by choosing its accurate parameters. One of

the global optimization techniques like PSO can be very

constructive to search out the optimized value of RBNN

parameters. PSO, developed by Kennedy and Elbert, in 1995 [12]

is based on the simulation of simplified animal social behavior

such as fish schooling, bird flocking etc.. Stochastic based search

algorithm PSO is a global searching technique with simplicity and

practicability and has been widely used in recent years to get the

optimal solutions [13].

Henceforth, in this paper, to develop the proposed hybrid

controller two important techniques naming Particle Swarm

Optimization (PSO) and the Neural Network (NN) have been

combined. This type of control schemes, taking advantageous

features of both the above mentioned PSO and NN intelligent

techniques and is named as Evolutionary Neural Networks

(ENN). By choosing PSO, auto adaptability quality is developed

in the RBNN [14]. In [15-16] such adaptive hybrid controllers

have been shown better control performance as compared to other

prevailing controllers. ENN has been called as the next generation

Neural Networks [17]. Moreover, some improvements in PSO

further add on performance quality as, Cao et al. [18] and Shi et

al. [19] used modified PSO to optimize RBNN and obtained

effective results.

Robotic manipulator is a highly non-linear, time-varying and

highly coupled system. For a manipulator, almost all kinds of

control techniques naming classical PD, PID, SMC, NN, etc. have

been compiled in literature [20, 21 and references there in]. But

because of the presence of the various structured and unstructured

uncertainties in the model dynamics; still the thrust for a perfect

and accurate controller is there.

In this paper, controller used is the hybrid of two model free

control techniques naming, PSO and NN. PSO is used to get the

finest possible performance deciding RBNN constants, naming

spread factor ( σj ) and weights of output layer (wjk). Thus, a

successful attempt to make a controller with great control outputs

for a manipulator has been made. For further improvement in the

control scheme, inertia weight factor of PSO is made adaptive.

For simulation purpose, a 2-DOF robotic manipulator having

planar elbow with remotely driven links manipulator has been

taken here. This type of model is with gear, linear, well

understood as the non-linear coupling between the motors has

been reduced. On the other hand, this gear introduces friction,

compliance, backlash in the dynamics. It has been observed from

the literature survey that a very few controllers has been

implemented for trajectory tracking control of this planar elbow

with remotely driven links manipulator. Performance of the

controller with this manipulator has been checked in presence of

payload mass changes and the unavoidable friction.

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Furthermore, the paper is organized as follows: Section II

deals with manipulator dynamics and the fundamentals of the

controllers; next, Section III contains the basic scheme of the

proposed controller of the paper. Simulation example and results

are given in Section IV. Finally, conclusions have been complied

in Section V.

2. Fundamentals

This section of the paper contains a brief review of the

manipulator dynamics and the intelligent techniques naming,

RBNN and PSO.

2.1. Manipulator Dynamics:

Fig. 2: Generalized coordinates for planar elbow manipulator with

remotely driven links

The excitation values of this Gaussian function are

distributed between the input values. The output of the hidden

layer is given by equation (2) as

The dynamics of revolute joint type of robot can be

described by following nonlinear Lagrange equation (1) [22], n

j=1 j

exp [

−ǁs−c

]

σj 2 (2)

M(q)q¨ + V(q, q˙) + G(q) = τ (1)

with q є Rn as the joint position variables, τ as vector of input

torques, M (q) is the symmetric and positive definite inertia

matrix, V(q, q˙) is the coriolis and centripetal matrix, G(q)

includes the gravitational forces. Input torque given to the

manipulator is of pivotal significance.

Manipulator used in this work is a planar elbow manipulator

with remotely driven link. Unlike planar elbow manipulator, in

this type of manipulator both the joints are driven by motors

mounted at the base. The first joint is turned directly by one of the

motors, while other is turned via a gearing mechanism or a timing

belt as in Fig 1. Here, the generalized coordinates taken are as in

Fig. 2, as the angle 𝑝2 is determined by driving motor number 2

and is not affected by the angle 𝑝1.

2.2 Radial Bias Neural Network (RBNN)

A typical RBNN consists of input layer, hidden layer and

output layer as represented if Fig [3]. Input layer consists of input

signals; hidden layer consists of radial bias functions (Gaussian

function); output layer gives output by multiplying weights with

the output of hidden layer. In this paper, input given to the RBNN

is error and velocity error (e and e˙) and output is obtained from

NN is the input torque to be given to the manipulator for

trajectory tracking control purpose.

Fig. 1: Two link revolute joint arm with remotely driven link

where j is the jth neuron of the hidden layer,

cj is the central position of the neuron j,

σj is the spread factor of Gaussian function.

In output layer, output vector is given by y = [τ1 τ2]T which

vectorily can be written as the output of kth neuron is given by

equation (3) 𝑛

= ∑ 𝑤

𝑗𝑘

∗ u

𝑗 =1

k 1,2 … . number of hidden layer neurons (3)

where wjk represents the linking weight of the neuron in the

output layer.

Fig. 3: RBNN architecture

Significance of the RBNN parameters to be optimized:

This section covers a brief discussion about the significance

of the spread factor ( ) and the output weights (wjk) in RBNN,

followed by a discussion on the proposed control scheme.

a. Spread factor ( ) is the first parameter to be optimized

using PSO. Spread factor () is of vital significance in

RBNN. Its too small value can result in a solution that

does not generalize from the input/target vectors and

with a large value of it, the radial basis neurons will

output large values (near 1.0) for all the inputs used to

design the network. If radial basis neurons always

output 1, any information presented to the network as

input becomes lost. Hence, it is required to choose

spread factor larger than the distance between adjacent

𝑝

u = ∑

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Neha Khurana

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input vectors, so as to get good generalization, but

smaller than the distance across the whole input space. It

can be assumed that, it is crucial to have accurate results

with the optimal value of this spread factor

b. Another performance deciding factor of RBNN is the

selection of output weights (wjk). Generally, these

weights from hidden to output layer are decided by

Least Square (LS) estimation [23]. In RBNN, these

output weights could be affected by very commonly

occurring noise and outliers in a nonlinear function.

Hence, the approximation precision of RBNN could be

consequently damaged with the presence of this external

noise and outliers in the data set. Hence, it is always

required to use some optimization technique to get the

values of these weights for the improvements in the

results and accuracy of NN based controllers.

2.3 Particle Swarm Intelligence (PSO)

In PSO starting with random population in search space, it

results in the optimal solution. During each step every particle is

accelerated towards its best neighboring position as well as in the

direction of global best position. Calculation of new position of

the swarm is given by equations (4) & (5) [12].

vid = vid + c1 ∈1 (pid − xid ) + c2 ∈2 (pxd − xid )

(4)

= x

+ v

(5)

where, in a D-dimensional space x

→

= (x

, x

,… x

) is a present

position vector, p

→

= (p

, p

, … p

) is a best position vector,

v

→

= (v

, v

, … v

) is a velocity vector, , c

and c

are constant

acceleration coefficients 2, ∈1 and ∈2 are the random number

generators. In [24, 25] it has been proved that PSO finds the

global best solution. PSO is becoming popular due to its

simplicity in implementation and ability to converge quickly to a

reasonably good solution.

Adaptive Weights in PSO

Although PSO is a new efficient emerging algorithm to the

family of evolutionary algorithms and proven to be better than

many other classical evolutionary techniques available (like

Genetic Algorithm (GA)), yet there lies a huge scope for multi

dimensional improvement in the basic PSO algorithm. One such

improvement is made by incorporating a weight parameter on the

previous velocity of the particle. The resulting equations for the

manipulation of the swarm are [26] given in equations (6) & (7)

vid = w ∗ vid + c1 ∈1 (pid − xid ) + c2 ∈2 (pxd − xid )

(6)

= x

+ v

(7)

where w is the inertia weight which manipulates the effects of the

previous velocities on the current velocity. It can be said that w

resolves the tradeoff between the global and the local exploration

ability of the swarm. Literature reveals that w should have greater

value in starting and should decrease gradually with iterations. As

suggested by Hou in 2008[27] w adjusted adaptively proves itself

as given in equation (8).

where a = 0.6, b = 1, iter is the current iteration.

This proposed adaptive weight in PSO has been applied to

the manipulator of a planar robot with remotely driven links for

the first time. Here, in this work i.e. for trajectory tracking control

of robotic manipulator, this adaptive PSO has proven itself.

2.4 Friction Modeling

Friction forces between two surfaces in contact arises as a

consequence of the irregularities and asperities at microscopical

level, and their effects depend on many factors, such as

displacement and relative velocity of bodies, properties of the

surface materials, presence of lubrication, temperature etc. The

experimental observation of friction phenomenon has led to

various, deeply different models, which capture the friction

component in a more or less accurate way. Friction is very

important for the control engineer. Friction should be as much as

reduced by good hardware design. But, with the advancements in

the computers, computer control has also shown the possibility to

reduce the effects of friction. This has been made possible using

various mathematical friction modes. Interesting reviews of the

main friction characteristics and classical models starting from

the basic concept of friction as a force that opposes motion,

captured by pure Coloumb model, up to complex static and

dynamic models like LuGre friction model has been provided in

literature. As opposed to classical static friction model, dynamic

friction models attempt to incorporate a variety of other friction

characteristics such as stiction, zero slip displacement, stribeck

effect etc. Dynamic friction models also tend to capture

effectively the changing friction characteristics that are caused

primarily due to wear and aging. One of the most accurate

dynamic frictions proposed is LuGre friction model. LuGre

Fiction can be modeled mathematically as in equations (9)

F = σoz + σ1z˙ + σ2v

z˙ = v − |v| z (9)

g(v)

g(v) = F

+ (F

− F

)exp⁡(

)

where z is average bristle deflection, σo is stiffness of bristles, σ1

is bristle damping coefficient, σ2 is viscous damping coefficient,

v is relative velocity between moving parts, Fc is coulomb

coefficient, Fs is static coefficient, vs is striberk velocity.

3. Proposed Controller

Even input output mapping in NN can be made by one of the

many possible mapping functions yet the key issue in RBNN is

not the selection of non-linear function but the key factor is the

selection of constant parameters of these non-linear functions.

Improper selection of some of the factors of RBNN can lead to

unsatisfactory control results from RBNN. Spread factor (σj) and

the network output weights (wjk) are the few most performance

deciding factors of RBNN. In other words, it can be said that

proper selection of spread factor ( σj ) and the network output

weights (wjk) can be adjusted using one of the upcoming latest

swarm intelligent technique naming PSO.

w =

b+[1g∗iter ]

(8)

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Volume 21, 2022

1 c1

2 1

Compare fitness evaluation with

population’s overall previous best

for finding global best.

Calculate new updated velocity

and position of the particles.

the total length; lci is the distance from the joint to the centre of

gravity; g is gravitational constant; τi is the torque input; where i

is 1 & 2 for link 1 & link 2. Parameters for the manipulator taken

for trajectory tracking are:

m1 = 10 ; m2 = 5 ; I1 = 0.2 ; I2 = 0.2 ; lc1 = 0.25; lc2 = 0.5 ;

g = 9.8.

Fig. 4: Working of the proposed control scheme

In Fig. 4, it can be seen that PSO is used to obtain the

optimized values of the RBNN parameters. PSO is initialized

using a random population. Adaptive inertia weight in PSO has a

different value for each iteration and hence, changes to adapt

itself to the running PSO. As fitness function is a part of the basic

PSO algorithm hence, it is evaluated for each iteration. Output of

this PSO (optimized spread factor and output weights) is provided

to RBNN.

This manipulator is made to track the path for a two-link

manipulator given by equation (12)

q1 = sin(0.67t) + sin(0.3t) (12a)

q2 = sin(0.39t) + sin(0.5t) (12b)

Table 1: PSO Parameters

Population size

Number of Iterations

Inertia Weight (w)

Acceleration factors (c1,c2)

Fitness function

Root mean square of tracking error

(RMSE)

This RBNN (with optimized constant parameters) is used to find

the control input torque to be given to the manipulator for

trajectory tracking. Tracking error and velocity tracking error are

the inputs to the RBNN to have the control torque (given to the

system to be controlled) as output. To have the values of error

and velocity error actual trajectory tracked is compared to the

desired trajectory. Flowchart representing the working of the

control system is given in Fig. 5.

4. Simulation Example and Results

For the verification of the proposed controller, in this section a

simulation study has been carried out. Control for a 2 DOF planar

elbow with remotely driven links has been using the proposed

controller has been implemented here. Dynamic model of the

manipulator has been given in equations (10), (11) [22]

d¨ 11 p1¨ + d¨ 12 p2¨ + c221 p2˙ + ∅1 = τ1 (10)

d¨ 21 p1¨ + d¨ 22 p2¨ + c112 p2˙ + ∅2 = τ2 (11)

where

d11 = m l2 + m l2 + I

= m

cos(p

− p

)

No Criterion met or

max. no. of iter?

Yes

Stop to get the

optimized values

of spread factor

d21

= m

cos(p2 − p1) and output

weights.

d22 = m l2 + I

2 c2 2

221

= −m

sin(p

− p

)

112

= m

sin(p

− p

)

g1 = (m1lc1 + m2l1)g cos(p1)

∅2 = m2lc2g cos(p2)

subscripts 1 & 2 indicates the link 1 & link 2; pi is the angle with

respect to horizontal axis; mi is the weight; Ii is the inertia; li is

Fig. 5: Flowchart representing control scheme for the proposed

controller

In this simulation study, for the trajectory tracking problem

of planar manipulator, various controllers discussed in this paper

have been implemented and the results have been compared. For

payload changes (m2 + ∆m) is taken as 1.35 kg i.e. 35 % rise in

Compare particle’s fitness

evaluation with pbest.

Start

Initialize spread factor (s) and

output weights randomly.

Evaluate Fitness Function.

Trained RBNN.

Inertia

Weight

Iterations

Initialize

Random

Population

Fitness

Function

Optimized Parameters

Desired

Actual

PSO

Adaptive

Weight

Manipulator

System

RBNN

Controller

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the mass of joint 2 of the manipulator. Parameters for LuGre

friction model are chosen as:

σo = 0.6, σ1 = 0.009, σ2 = 0.6, Fs = 0.01, Fc = 10

4.1 RBNN Controller

For the implementation of the RBNN controller, given in

Section II, the spread factor has been tuned manually between 0-

2. The performance of the RBNN has been found best with spread

factor as 2.

4.2 PSO Controller

Parameters chosen for a basic PSO controller have been

given in Table 1 and results have been compiled in Fig. [6-9].

4.3 Proposed Controller

In proposed controller parameters for PSO are given in table

1 except the weight factor w, which is the adaptive in nature and

is given in (8). Search range for spread factor has been taken as

(0-2) and range for output weight factors has been taken as (0-1).

Results for the trajectory tracking by the manipulator have been

presented Figs. [6-9]. Although the graphs in Figs. [6-7] presents

that the trajectory tracked by the manipulator using different

control schemes is very close to each other, but the control

performance of controllers can be easily differentiated with the

help of tracking error graphs plotted in Figs. [8-9]. These graphs

clearly represent that the best tracking performance is given by

the proposed controller. Tracking errors of various controllers are

given in Figs. [8-9]. Table 2 contains the performance indices to

evaluate the performance of the controllers with all the

uncertainties in terms of mean and mean square error (mse).

Other type of error measuring performance indices like 2-norm

error, integral square error (ISE) can also be evaluated and are

found to show the similar type of results. It has been observed

from table 2 that the max and mean error in Joint 1 & 2 is about

100 times lesser that the max and mean error of the other

controllers. In joint 1, it can be observed that the mse is about 104

times lesser than the mse in other two control schemes whereas in

joint 2 mse in the proposed controller is about 103 times lesser

than the mse in other two control schemes. Hence, along with the

robustness in the proposed control scheme, there is a rise in the

accuracy in the tracking performance of the system under study.

Execution time (in seconds) for each control technique has been

tabulated in table 3. It has been observed that RBNN is taking the

maximum time for control execution. Proposed controller, along

with less tracking error, is implementing the control action in

lesser time when compared with RBNN.

5. Conclusion

As said, it would be safe here to infer again that the most

commonly and widely used neural networks (NN) are not

flawless, rather they have various shortcomings of their own

including the dependency on experts for tunings its parameters,

such as spread factor and output weights for good accuracy in

results. This need is fulfilled by the proposed controller which

uses one of the most emerging optimization techniques named as

particle swarm optimization (PSO) to get the optimized

parameters of RBNN for enhanced performance. This PSO

enhanced RBNN controller has proved itself with accuracy in

trajectory tracking. This controller also converges itself in lesser

time as compared to a simple RBNN controller. Hence, as the

outcome of the paper, it can be said that with the proposed robust

control scheme perfect trajectory tracking problem of robotic

manipulator has been solved upto a mark.

The study opens new vistas and futuristic avenues for

further study, the more advanced and upgraded versions of PSO

may be used for optimizing RBNN.

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Neha Khurana

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Table 2: Tracking Errors: Joint 1 & 2

Uncertainties

Control

Scheme

Joint 1

Joint 2

max.

abs. error

Mean

error

mse

max.

abs. error

Mean

error

mse

Payload

changes

RBNN

0.0494

0.0326

0.0013

0.0686

0.0337

0.0014

PSO

0.0613

2.15e-02

7.51e-04

0.0545

2.24e-02

7.00e-04

Proposed

0.002

0.0018

3.32e-06

4.13e-04

2.56e-04

6.69e-08

LuGre Friction

RBNN

0.116

0.0318

0.002

1.03e-01

0.0306

0.0018

PSO

0.0697

0.0199

7.57e-04

0.0587

0.0173

5.70e-04

Proposed

2.53e-04

7.80e-05

1.75e-08

2.92e-04

1.77e-04

3.44e-08

Both

RBNN

0.0587

0.0257

9.18e-04

0.0618

0.0248

9.15e-04

PSO

0.0637

0.0236

8.35e-04

0.0505

0.0197

5.72e-04

Proposed

4.31e-04

2.55e-04

7.58e-08

4.52e-04

3.81e-04

1.47e-07

Table 3: Control execution time (in seconds)

Controllers

RBNN

PSO

Proposed

Controller

time

101.99

28.77

48.34

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS

DOI: 10.37394/23201.2022.21.13

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author 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 author has no conflict of interest to declare that

is relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

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Fig. 6: Trajectory tracking response by Joint 1

Fig. 7: Trajectory tracking response by Joint 2

Fig. 8: Tracking error for Joint 1

Fig. 9: Tracking error for Joint 2

WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS

DOI: 10.37394/23201.2022.21.13

Neha Khurana

E-ISSN: 2224-266X

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Volume 21, 2022