DC MOTOR USING MULTI ACTIVATION WAVELET

NETWORK (MAWN) AS AN ALTERNATIVE TO

A PD CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

Walid Emar, Noora Khalaf, Maher Dababneh and Waleed Johar

Al-Isra Private University, Amman Jordan

Key words: Robotics, Control System, Wavelet Network, DC motor controller

.

Abstract: In this paper, a robust MAWN is proposed. An application that constructs Wavelet Network as an

alternative to a PD controller in the robotics control system with DC motor is fully investigated.

Experimental results not only show that the target performance can be achieved by the proposed Wavelet

Network, but also it outperforms the conventional PD controller. An literature survey was conducted to shed

some light into this research field shows a sparsity of work addressing this concept, and this what stimulated

the idea of this work.

1 INTRODUCTION

The design of intelligent, autonomous machines to

perform tasks that are dull, repetitive, hazardous, or

that require skill, strength, or dexterity beyond the

capability of humans is the ultimate goal of robotics

research. Examples of such tasks include

manufacturing, excavation, construction, undersea,

space, and planetary exploration, toxic waste

cleanup, and robotic assisted surgery. Robotics

research is highly interdisciplinary requiring the

integration of control theory with mechanics,

electronics, artificial intelligence and sensor

technology (Xiao, 2001).

The ever increasing technological demands of

to

day, call for very complex systems, which in turn

require highly sophisticated controllers to ensure

that high performance can be achieved and

maintained under adverse conditions. There are

needs in the control of these complex systems,

which cannot be met by conventional approaches to

control. For instance, there is a significant need to

achieve higher degrees of autonomous operation for

robotic systems, spacecraft, manufacturing systems,

automotive systems, underwater and land vehicles,

and others. To achieve such highly autonomous

behavior for complex systems, one can enhance

today's control methods using intelligent control

systems and techniques (Feitosa et al., 2000).

Intelligent control methodologies are being

appl

ied to robotics and automation, communications,

manufacturing, traffic control. To mention few

application areas: neural networks, fuzzy control,

genetic algorithms, planning systems, expert

systems, and hybrid systems are all related areas.

The term "intelligent control" has come to mean,

particularly to those outside the control area, some

form of control using fuzzy and/or neural network

methodologies (Sgarbiy et al., 1997).

Neural networks have been applied very

success

fully in the identification and control of

dynamic systems. The universal approximation

capabilities of the multilayer perceptron (the

backpropogation algorithm) make it a popular

choice for modeling nonlinear systems and for

implementing general-purpose nonlinear controllers

(Calise and Rysdyk, 1996). The combination of soft

computing and wavelet theory has lead to a number

of new techniques: MAWN, wavenets, and fuzzy-

wavelet (Yao, 1999).

It is difficult to model the environment to

p

rovide the controller with the relevant data and

program actions for all possible situations. Hence,

controllers with abilities to learn and to adapt are

needed to solve this problem. Soft computing

provides an attractive venue to deal with these

situations. Soft computing methods are based on

biological systems and can provide the following

features: generalization, adaptation and learning. As

more is realized about the use and properties of soft

computing methods, the development of controller is

341351

Emar W., Khalaf N., Dababneh M. and Johar W. (2007).

DC MOTOR USING MULTI ACTIVATION WAVELET NETWORK (MAWN) AS AN ALTERNATIVE TO A PD CONTROLLER IN THE ROBOTICS

CONTROL SYSTEM.

In Proceedings of the Second International Conference on Signal Processing and Multimedia Applications, pages 341-347

DOI: 10.5220/0002131603410347

Copyright

c

SciTePress

shifting towards using soft computing (Gu and Hu,

2002).

In this paper a robust Multi Activation Function

Wavelet Network Wavelet (MAFWN) is used as a

controller analogous to a PD controller in the control

of a robotic arm and a payload system with a DC

motor that is required for conducting a pick and

place operation to achieve the required performance.

2 THE MULTI MAFWN

An application of multi wavelet filters to neural

networks is investigated in this paper. This new

technique called MAFWN. It is an interesting

alternative to wavelet networks that absorbs the

advantage of high resolution of wavelets and the

advantages of learning feed-forward neural

networks.

The MAFWN is very similar to wavelet Network

(WN) but, has some important differences, whereas

wavelets have an associated scaling function φ(t)

and wavelet function ψ(t). MAWN has multi scaling

φ1(t), φ2(t) … φn(t), and multi wavelet functions

ψ1(t), ψ2(t) … ψn(t). However, Two AFWN

(TAFWN) has two scaling functions φ1(t), φ2(t) and

two wavelet functions ψ1(t), ψ2(t). Subsequently,

there are two scaling filters and two wavelet filters

for the case of TAFWN, and this will be considered

as a case study for this research.

3 WAVELET NETWORK

ALGORITHM

The two activation function wavelet network

(TAFWN) architecture approximates any desired

signal by generalizing a linear combination of

two set of daughter wavelets and

)(ty

)(

,,1

th

ba

)(

,,2

t

h

ba

,

where the daughter wavelets

)(

,,1

t

h

ba

and

)(

,,2

t

h

ba

are generated by dilation, a , and

translation, , from two mother wavelets

b

)()(

21

τ

τ

handh

, where

a

bt −

=

τ

.

)()(

1

,,1

a

bt

hth

ba

−

=

(1)

)()(

2

,,2

a

bt

hth

ba

−

=

(2)

Where:

a

: Dilation factor, with

a

> 0.

b

: Translation factor.

:

t

Signal time interval

The network architecture is shown in figure (1).

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

11

11

11

a

bt

h

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

21

21

21

a

bt

h

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

12

12

12

a

bt

h

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

22

22

22

a

bt

h

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛−

k

k

k

a

bt

h

1

1

1

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛−

k

k

k

a

bt

h

2

2

2

X

(

t

)

)(ty

∧

W

11

W

12

W

22

W

21

W

1k

W

2k

Figure 1: Structure of TAFWN.

A TAFWN is a 3-layers feed forward neural

network. First the TAFWN parameters, dilation a's,

translation b's, and weight w's should be initialized,

and the desired sets of data, the input signal x(t), the

desired output (target) y(t), the number of scaling

functions p (p=2 in this work) and the number of

wavelons k are given. The approximated signal of

the network can be represented by equation:

)(

ˆ

ty

TAFWN is trained by the gradient descent

algorithms like least mean squares (LMS) to

minimize the mean-squared error. During learning,

the parameters of the network are optimized.

∑∑

×=

==

k

i

ij

b

ij

aij

p

j

thwtxty

1

,

,

,

,

1

)()()(

ˆ

(3)

Where: x(t) is the input signal.

ij

w

,

is the weight coefficients between hidden

and output layers.

j=1,2,…, p. p=2: a number of scaling functions.

i=1,2,…, k. k is a number of wavelons.

ij

b

ij

a

h

,

,

,

is a two set of daughter wavelets generated

from two mother wavelets

)(),(

21

thth

as in

equations (1) and (2) respectively.

The TAFWN parameters

i

,

j

w

,

ij

a

,

, and

i

can

be optimized in the LMS algorithm by minimizing a

cost function or the energy function,

j

b

,

E

, over all

function interval. The energy function is defined by

equations (4) and (5), is the desired output

(target) and is the actual output signal of

TAFWN.

)(ty

)(

ˆ

ty

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

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SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

352

∑

=

=

T

t

teE

1

2

)(

2

1

(4)

∑

=

−=

T

t

tytyE

1

2

))(

ˆ

)((

2

1

(5)

Where,

T

is the total interval of function,

)(

t

y

is the desired output (target) and is the actual

output signal of WN.

)(

ˆ

ty

To minimize then the method of steepest

descent is used, which requires the gradients

E

ij

w

E

,

∂

∂

,

ij

a

E

,

∂

∂

, and

ij

b

E

,

∂

∂

for updating the incremental changes to each

particular parameter

ij

w

,

,

ij

a

,

, and

ij

b

,

,

respectively. The gradients of are given as

follows:

E

() ( ) ()

txhte

w

E

T

t

ij

τ

∑

−=

∂

∂

=1

,

(6)

() ()

()

ij

b

h

ij

T

t

ij

wtxte

b

E

,

,

1

,

∂

∂

=

∑

−=

∂

∂

τ

(7)

ij

T

t

ij

ij

ij

b

E

b

h

wtxte

a

E

,

1

,

,

,

)(

)()(

∂

∂

=

∑

∂

∂

−

=

∂

∂

=

τ

τ

τ

(8)

ij

ij

a

bt

,

,

−

=

τ

(9)

Derivatives of the various wavelet filters with

respect to its translation

ij

b

h

,

)(

∂

∂

τ

, are given in (Oussar

et al., 1996).

The incremental changes of each coefficient are

simply the negative of their gradients.

w

E

w

∂

∂

−=Δ

(10)

b

E

b

∂

∂

−=Δ

(11)

a

E

a

∂

∂

−=Δ

(12)

Thus, each coefficient , and of the

network is updated in accordance with the rule

given:

w

b

a

w

t

w

t

w

w

Δ+=+

μ

)()1(

(13)

b

t

b

t

b

b

Δ+

=

+

μ

)()1(

(14)

atata

a

Δ+

=

+

μ

)()1(

(15)

Where,

μ

is the fixed learning rate parameter

(Oussar et al., 1996).

2. Set: the number of trainings, iter =0, the

incremental changes of each coefficient,

0),,(

=

Δ

Δ

Δ

baw

, and the initial square error,

5.0

=

ite

r

E

3. Calculate the approximated signal of the

network using equation (3).

)(

ˆ

ty

4. Calculate the gradients of each coefficient

using equations (6), (7), (8) and calculate the

coefficients incremental changes which are the

negative of their gradients.

5. Choose a constant

μ

, such that 0.01 ≤

μ

≤1

and calculate the new coefficients ,

1+iter

w

1+ite

r

b

, and

1+iter

a

of the network in accordance

with the rules given in equations (13), (14) and (15).

6. Calculate the square error

1+ite

r

E

using

equation (5).

If

1+ite

r

E

is small enough, then the training is

good and the run of the algorithm is stopped.

Otherwise, set iter = iter + 1 and go to (3) again.

At every iteration, the network parameters are

modified using the gradient descent algorithm that

will result in minimizing the parameter E.

The training algorithm of the proposed TAFWN

consists of the following six steps:

1. Initialize TAFWN parameters, dilation a's,

translation b's, and weight w's, p=2, two mother

wavelets filters

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

−−

i

a

i

bt

i

a

i

bt

hh

21

,

,

the desired sets of data, the input signal x(t), the

desired output (target) y(t), and the number of

wavelons k are given.

4 MAWN FOR CONTROLLING A

ROBOTIC ARM

An example-control of a robotic arm and a payload

system with a DC motor- is given in this paper to

illustrate the use of the proposed MAWN as a PD

controller and its performance.

DC MOTOR USING MULTI ACTIVATION WAVELET NETWORK (MAWN) AS AN ALTERNATIVE TO A PD

CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

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DC MOTOR USING MULTI ACTIVATION WAVELET NETWORK (MAWN) AS AN ALTERNATIVE TO A PD

CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

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4.1 Robotic Arm Properties

The Robotic arm is depicted in Figure (2). It is

composed of a rigid beam which is connected to a

motor shaft to create a robotic system conducting a

pick and place operation. A solid disk is attached to

the end of the beam through a magnetic device (e.g.,

a solenoid). If the magnet is on, the disk will stick to

the beam, and when the magnet is turned off, the

disk is released. The objective of the robotic arm is

to drop the disk into a hole as fast as possible. The

hole is 1 inch (25.4 mm) below the disk as shown in

figure (3) (Oussar and Dreyfus, 1996). The robot

arm is required to move in one direction only, from

the initial position. Also, the hole location may be

anywhere within an angular range of 20° to 180°

from the initial position. It is in the angular position

of 150° for the sake of this example. The idea of this

control system is to move a metal object attached to

a robot arm by an electromagnet from position 0° to

the angular position 150° with a specified overshoot

and minimum overall time.

Star

Motor

Pa

y

l

o

a

d

Dro

p

off tar

g

e

t

Figure 2: Control of a robotic arm and payload.

A system simulink model of the system is shown

in figure (4) and the simulated DC Motor is

portrayed in Figure (5). This figure represents a

simple PD controller model with proportional gain

of 15 and derivative gain of 2.1. In the

Electromagnet Control block, drop-off payload

location and the time delay (in seconds) to turn the

magnet off after reaching the target parameters is

adjusted to 150° and 0.8 sec, respectively. So, the

"Drop position angle" is the angle where the

electromagnets turn off, thereby, dropping the

payload." However, start to wait for drop position at

time" refers to the time where the position triggers

starts to wait for the position specified by "Drop

position angle." An overall time response for the

system with PD controller is shown in Figure (6).

Figure 3: The robotic arm system simulink model.

4.2 TAFWN as PD Controller

Now, the PD controller shown in figure (5) is

replaced with the proposed TAFWN structure.

TAFWN of 40 [Morlet, Rasp2] filters and fixed

learning rate of 0.1 is trained first with the desired

input-output data set shown in figure (6- a and b).

Figure (7) however, shows the training performance

of the network.

After training to MSE value less than e-005, the

trained TAFWN is employed to control the robotic

arm system, the system simulink model with

TAFWN controller is shown in figure (8). It is clear

from the above results that the TAFWN is proved to

be a PD controller, and its position response per time

Figure 4: The simulated DC Motor.

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Time (sec)

Figure 5: Position response (degree) per time (sec) with PD controller.

Time (sec)

(a) the input to TAFWN

Time (sec)

(b) the desired output of TAFWN

Figure 6: The desired input-output data set.

Figure 7: Mean-Square Error per learning iteration.

DC MOTOR USING MULTI ACTIVATION WAVELET NETWORK (MAWN) AS AN ALTERNATIVE TO A PD

CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

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DC MOTOR USING MULTI ACTIVATION WAVELET NETWORK (MAWN) AS AN ALTERNATIVE TO A PD

CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

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Figure 8: Robotic arm system simulink model with TAFWN controller.

Position (deg)

Time (sec)

Figure 9: Position response (degree) per time (sec) with TAFWN controller.

is illustrated in figure (9). As it is shown at the

specified time (0.8 sec), the angular position angle

150° and so the metal object attached to the robot

arm is gotten to be in the "Drop position angle"

(150°) at time (0.8 sec). Hence there is no significant

difference between the position responses for both

PD and TAFWN controllers.

5 CONCLUSIONS

In this paper, an advanced wavelet network, called

Two Activation Function Wavelet Network is

presented as an interesting alternative to wavelet

networks. This technique absorbs the advantage of

high resolution of wavelets and the advantages of

learning and feed-forward of neural networks. The

algorithm of function identification is designed and

implemented using Matlab 6.5 tool.

The Two Activation Function Wavelet Network

(TAFWN) structure is implemented and several

examples are carried out to verify this

implementation. It can be concluded that this

structure achieves an approximation assuming

reasonable choice of the number of wavelons and

mother wavelet basis functions. The Two Activation

Function Wavelet Network is proved to be a

controller analogous to PD controller. After the off-

line training of the TAFWN controller, it shows the

ability to get the specified position response exactly

at the specified time when it's embedded in the

control system. No significant difference between

the position responses for both PD and the proposed

TAFWN controllers, indicating further the validity

of the idea of this research.

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SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

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CONTROLLER IN THE ROBOTICS CONTROL SYSTEM

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