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ARTIFICIAL NEURAL NETWORK

CHAPTER 1

INTRODUCTION

1.1 Background

An artificial neural network is a system based on the operation of biological neural networks, in

other words, is an emulation of biological neural system. Why would be necessary the

implementation of artificial neural networks? Although computing these days is truly advanced,

there are certain tasks that a program made for a common microprocessor is unable to perform;

even so a software implementation of a neural network can be made with their advantages and

disadvantages.

1.2 Basics of ANN

This terminology has been developed from the biological model of the brain. A neural network

consists of a set of connected cells: THE NEURONS. The neurons receive impulses either from

input cells or other neurons and perform some kind of transformation on the input and transmit

the outcome to other neurons or output cells. The neural networks are built from layers of

neurons connected so that one layer receives an input from the receding layer of neurons andpasses the output on to the subsequent.

1.3 Style of Neural Computation

Figure 1.1 The Style Of Neural Computation

DEPARTMENT OF BIOTECHNOLOGY, SCOE, PUNE-1-


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An input is presented to the neural network and a corresponding desired or target response set at

the output (when this is the case the training is called supervision). An error is composed from

the difference between the desired response and the system output. This error information is fed

back to the system and adjusts the system parameters in a systematic fashion (the learning rule).

The process is repeated until the performance is acceptable. It is clear from this description that

the performance hinges heavily on the data. If one does not have data that cover a significant

portion of the operating conditions or if they are noisy, then neural network technology is

probably not the right solution. On the other hand, if there is plenty of data and the problem is

poorly understood to derive an approximate model, then neural network technology is a good

choice. In artificial neural networks, the designer chooses the network topology, the performance

function, the learning rule, and the criterion to stop the training phase, but the system

automatically adjusts the parameters. So, it is difficult to bring a priori information into the

design, and when the system does not work properly it is also hard to incrementally refine the

solution. But ANN-based solutions are extremely efficient in terms of development time and

resources, and in many difficult problems artificial neural networks provide performance that is

difficult to match with other technologies. Denker 10 years ago said that "artificial neural

networks are the second best way to implement a solution" motivated by the simplicity of their

design and because of their universality, only shadowed by the traditional design obtained by

studying the physics of the problem. At present, artificial neural networks are emerging as the

technology of choice for many applications, such as pattern recognition, prediction, system

identification, and control.



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Figure 1.2: Computation of error



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convey information to the brain. All these signals are identical. Therefore, the brain

determines what type of information is being received based on the path that the signal

took. The brain analyzes the patterns of signals being sent and from that information it can

interpret the type of information being received. This ensures that the signal traveling

down the axon travels fast and remains constant (i.e. very short propagation delay and no

weakening of the signal). The synapse is the area of contact between two neurons. The

neurons do not actually physically touch. They are separated by the synaptic cleft. The

neuron sending the signal is called the pre -synaptic cell and the neuron receiving the

signal is post -synaptic cell.

2.2 The Mathematical Model

When creating a functional model of the biological neuron, there are three basic components of

importance. First, the synapses of the neuron are modeled as weights. The strength of the

connection between an input and a neuron is noted by the value of the weight. Negative weight

values reflect inhibitory connections, while positive values designate excitatory connections. The

next two components model the actual activity within the neuron cell. An adder sums up all the

inputs modified by their respective weights. This activity is referred to as linear combination.

Finally, an activation function controls the amplitude of the output of the neuron. An acceptable

range of output is usually between 0 and 1, or -1 and 1. Mathematically, this process is described

in the figure.



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Figure 2.2: Mathematical computation for ANN

From this model the interval activity of the neuron can be shown to be:

The output of the neuron, yk, would therefore be the outcome of some activation function on the

value of vk.

2.3 Activation Functions

As mentioned previously, the activation function acts as a squashing function, such that the

output of a neuron in a neural network is between certain values (usually 0 and 1, or -1 and 1). In

general, there are three types of activation functions, denoted by (.) . First, there is the

Threshold Function which takes on a value of 0 if the summed input is less than a certain



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threshold value (v), and the value 1 if the summed input is greater than or equal to the threshold

value.

Secondly, there is the Piecewise-Linear function. This function again can take on the values of 0

or 1, but can also take on values between that depending on the amplification factor in a certain

region of linear operation.

Thirdly, there is the sigmoid function. This function can range between 0 and 1, but it is also

sometimes useful to use the -1 to 1 range. An example of the sigmoid function is the hyperbolic

tangent function.



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Figure 2.3: Graphs of activation functions

2.4 Kinematics

Kinematics is the science of motion which treats motion without regard to the forces which cause

it. Within this science one studies the position, velocity, acceleration, and all higher order

derivatives of the position variables. A very basic problem in the study of mechanical

manipulation is that of forward kinematics. This is the static geometrical problem of computing

the position and orientation of the end-effector ('hand') of the manipulator. Specifically, given a

set of joint angles, the forward kinematic problem is to compute the position and orientation of

the tool frame relative to the base frame.



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Figure 2.4:An exemplar robot manipulator

2.5 Inverse Kinematics

This problem is posed as follows: given the position and orientation of the end-effector of the

manipulator, calculate all possible sets of joint angles which could be used to attain this given

position and orientation. This is a fundamental problem in the practical use of manipulators. The

inverse kinematic problem is not as simple as the forward one. Because the kinematic equations

are nonlinear, their solution is not always easy or even possible in a closed form. Also, the

questions of existence of a solution, and of multiple solutions, arise.

2.6 Dynamics

Dynamics is a field of study devoted to studying the forces required to cause

motion. In order to accelerate a manipulator from rest, glide at a constant end-effector velocity,

and finally decelerate to a stop, a complex set of torque functions must be applied by the joint

actuators. In dynamics not only the geometrical properties (kinematics) are used, but also the



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physical properties of the robot are taken into account. Take for instance the weight (inertia) of

the robot arm, which determines the force required to change the motion of the arm. The

dynamics introduces two extra problems to the kinematic problems.

1) The robot arm has a 'memory'. Its responds to a control signal depends also on its history (e.g.

previous positions, speed, acceleration).

2) If a robot grabs an object then the dynamics change but the kinematics do not change. This is

because the weight of the object has to be added to the weight of the arm (that's why robot arms

are so heavy, making the relative weight change very small).

2.7 End-Effector Positioning

The main aim in robot manipulator control is often the positioning of the hand or end-effector in

order to be able to, e.g., pick up an object. With the accurate robot arms that are manufactured,

this task is often relatively simple, involving the following steps:

1. Determine the target coordinates relative to the base of the robot. Typically, when this

position is not always the same, this is done with a number of fixed cameras or other

sensors which observe the work scene, from the image frame determine the position of

the object in that frame, and perform a pre-determined coordinate transformation;

2. With a precise model of the robot (supplied by the manufacturer), calculate the joint

angles to reach the target (i.e., the inverse kinematics).

3. Move the arm (dynamics control) and close the gripper.

2.8 Need For Involvement Of Neural Networks

So if these parts are relatively simple to solve with a high accuracy, why involve neural

networks? The reason is the applicability of robots. When 'traditional' methods are used to

control a robot arm, accurate models of the sensors and manipulators (in some cases with

unknown parameters which have to be estimated from the system's behavior ; yet still with

accurate models as starting point) are required and the system must be calibrated. Also, systems

which suffer from wear-and-tear (and which mechanical systems don't?) need frequent



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recalibration or parameter determination. Finally, the development of more complex (adaptive!)

control methods allows the design and use of more flexible (i.e., less rigid) robot systems.

2.9 Camera Robot Co-Ordination

The system we focus on in this section is a work door observed by fixed cameras, and a robot

arm. The visual system must identify the target as well as determine the visual position of the

end-effectors.

The target position Xtarget together with the visual position of the hand Xhand are input to

the neural controllerN(.). This controller then generates a joint position for the robot:

=N(Xtarget ,Xhand)

We can compare the neurally generated with the optimal 0 generated by a fictitious perfect

controllerR(.):

0 = R(Xtarget,Xhand):

The task of learning is to make the N generate an output 'close enough' to 0 There are two

problems associated with teachingN(.):

1. Generating learning samples which are in accordance with eq. 0 = R(Xtarget,Xhand).

This is not trivial, since in useful applications R(.) is an unknown function. Instead, a

form of self-supervised or unsupervised learning is required.

2. Constructing the mapping N(.) from the available learning samples. When the (usually

randomly drawn) learning samples are available, a neural network uses these samples to

represent the whole input space over which the robot is active. This is evidently a form ofinterpolation, but has the problem that the input space is of a high dimensionality, and the

samples are randomly distributed.

2.10 Feed Forward Networks



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When using a feed-forward system for controlling the manipulator, a self-supervised learning

system must be used. One such a system has been reported by (Psaltis, Sideris, & Yamamura,

1988). Here, the network, which is constrained to two-dimensional positioning of the robot arm,

learns by experimentation. Three methods are proposed:

1. Indirect learning: In indirect learning, a Cartesian target point x in world coordinates is

generated, e.g., by a two cameras looking at an object. This target point is fed into the

network, which generates an angle vector . The manipulator moves to position , and the

cameras determine the new position x' of the end-effector in world coordinates. This x'

again is input to the network, resulting in '. The network is then trained on the error

1= -'

Figure 2.5: ANN indirect learning model

2. Indirect learning system for robotics. In each cycle, the network is used in two different

places: first in the forward step, then for feeding back the error.

However, minimization of 1 does not guarantee minimization of the overall error = x-

x'. For example, the network often settles at a 'solution' that maps all x's to a single .

3. General learning: Here the plant input must be provided by the user. Thus the network

can directly minimize | - '|. The success of this method depends on the interpolation

capabilities of the network.

4. Specialized learning: Keep in mind that the goal of the training of the network is to

minimize the error at the output of the plant: = x - x'. We can also train the neural



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network by 'back propagating' this error trough the plant (compare this with the back

propagation of the error). This method requires knowledge of the Jacobian matrix of the

plant. A Jacobian matrix of a multidimensional function F is a matrix of partial

derivatives of F, i.e., the multidimensional form of the derivative. For example, if we

have

Y=F(X),

then

or



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or

where J is the Jacobian matrix of F. So, the Jacobian matrix can be used to calculate the

change in the function when its parameters change. Now, in this case we have

where Pi( ) the ith element of the plant output for input . The total error = x - x' is

propagated back through the plant by calculating the as in eq.

where i iterates over the outputs of the plant. When the plant is an unknown function,

where can be approximated by

where ej is used to change the scalar j into a vector. This approximate derivative can be

measured by slightly changing the input to the plant and measuring the changes in the

output.

Again a two-layer feed-forward network is trained with back-propagation. However, instead of

calculating a desired output vector the input vector which should have invoked the current output

vector is reconstructed, and back-propagation is applied to this new input vector and the existing

output vector. The configuration used consists of a monocular (with 1 eye) manipulator which

has to grasp objects. Due to the fact that the camera is situated in the hand of the robot, the task

is to move the hand such that the object is in the centre of the image and has some predetermined



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size in which the visual flow-field is used to account for the monocularity of the system, such

that the dimensions of the object need not to be known anymore to the system. One step towards

the target consists of the following operations:

1. Measure the distance from the current position to the target position in camera domain, x.

2. Use this distance, together with the current state of the robot, as input for the neural

network. The network then generates a joint displacement vector .

3. Send to the manipulator.

4. Again measure the distance from the current position to the target position in camera

domain, x'.

5. Calculate the move made by the manipulator in visual domain, where is

the rotation matrix of the second camera image with respect to the rest camera image;

6. Teach the learning pair to the neural network.

This system has shown to learn correct behavior in only tens of iterations, and to be very

adaptive to changes in the sensor or manipulator.

By using a feed-forward network, the available learning samples are approximated by a single,

smooth function consisting of a summation of sigmoid functions. A feed-forward network with

one layer of sigmoid units is capable of representing practically any function. But how are the

optimal weights determined in finite time to obtain this optimal representation? Experiments

have shown that, although a reasonable representation can be obtained in a short period of time,

an accurate representation of the function that governs the learning samples is often not feasible

or extremely difficult. The reason for this is the global character of the approximation obtained

with a feed-forward network with sigmoid units: every weight in the network has a global effect

on the final approximation that is obtained. Building local representations is the obvious way

out: every part of the network is responsible for a small subspace of the total input space. Thus

accuracy is obtained locally (Keep It Small & Simple). This is typically obtained with a

Kohonean neural network.



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Figure 3.1 Three degree of freedom robot manipulator with two revolute

joints and one prismatic joint.

Assume that there is a coordinate system for the robot manipulator as shown

at figure 2.



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Figure 3.2: Co-ordinate system of the robot manipulator.

The position of point P(x ,y) can be determined by using following equation:

P(x,y) = f(1, 2)

(2)

Assume that vector P is determined by adding vector r1 and vector r2 where

r1 and r2 can be defined as follows:

r1 = [l1 cos1 , l1 sin1]

(3)

r2 = [l2 cos(1 +2) , l2 sin(1 +2)]

(4)

Then

x= l1cos1 + l2 cos (1 +2)

(5)

y= l1sin1 + l2 sin (1 +2)

(6)

Equation (5) and (6) are forward kinematics equation of the robot

manipulator with two revolute joints.

The inverse kinematics equation can be derived from (5) and (6). By using

trigonometry equation, (5) and (6) can be written as follows:

x= l1cos1 + l2 cos1cos 2 l2 sin1sin2

(7)

y= l1sin1 + l2 sin1cos 2 + l2 cos1sin2(8)

Solving (7) and (8) to find value of1 and 2 results the following equation:



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1=tan-1((y(l1+ l2 cos2)-x l2 sin2 ) / (x(l1+l2 cos2) +y l2 sin2 ))

(9)

2=cos-1((x*x +y*y-l1*l1-l2*l2) / (2 * l1 * l2))

(10)

Equation (9) and (10) are inverse kinematics equation of the robot

manipulator, which has two revolute joints. For a set of given coordinate P(x ,

y), the position of each link (1, 2) can be calculated by using (9) and (10).

3.2 Modeling Inverse Kinematics Using Artificial Neural Network

Here artificial neural network was applied for modeling the inverse kinematics of the robot

manipulator and the model is only considered for the kinematics of two revolute joints. The

architecture of artificial neural network used in this research is a multi layer perceptron with

steepest descent back propagation training algorithm. Figure 3 shows the architecture of artificial

neural network used for modeling inverse kinematic of the robot manipulator.



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Figure 3.3: Layers of ANN

The network has two input neurons. Both input neurons receive the information of tool

coordinateP(x, y). The output layer of the network has two neurons. Both neurons in the outputlayer give information about the position of each link (1, 2). Number of hidden layer of the

network varies from 1 to 10 layers and number of neuron per hidden layer varies from 1 to 50

neurons. If the artificial neural network has m layers and receives input of vectorp, then theoutput of the network can be calculated by using the following equation:

(a^m) = (f^m)(W^m*f m-1(W^m-1*f^m-2(.W^2*f(w^1*p+b^1)+b^2)+b^m-1)+b^m)

(11)

Wherefm is log-sigmoid transfer function of the mth layer of the network that can be defined as

following equation:

F( n)= (1+(e^(-n)))^(-1) (12)DEPARTMENT OF BIOTECHNOLOGY, SCOE, PUNE

-20-


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Wm is weight of the mth layer of the network, and b m is bias of the mth layer of the network.

Equation (11) is known as the feed forward calculation. Back propagation algorithm is used as

the training method of the designed artificial neural network. The back propagation algorithm

includes the following steps:

1. Initialize weights and biases to small random numbers.

2. Present a training data to neural network and calculate the output by propagating the input

forward through the network using (11).

3. Propagate the sensitivities backward through the network:

S^m = (-2)*F^m*(n^m)*(t-a)

(13)

S^m = F^m * (n^m)*(W^(m+1))^t * (S^m+1)

(14)

4. Calculate weight and bias updates

(W ^m)(k) = - * (S^m) *( ^m-1)^t

(15)

(b^m)^(k) = - *(S^m)(16)

Where is learning rate.

5. Update the weights and biases

(W ^m)(k+1) = (W ^m) + (W ^m)(k)

(17)

(b ^m)(k+1) = (b ^m) + (b ^m)(k)

(18)

6. Repeat step 2 5 until error is zero or less than a limit value.



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3.3 Experimental Result

Experiment was done to see performance of artificial neural network in order to model the

inverse kinematic of the robot manipulator. The performance of artificial neural network is

indicated by the RMSE (ROOT MEAN SQUARE ERROR) value. Experiment was done invarious training parameter value of artificial neural network, i.e., various numbers of hidden

layers, various number of neuron per layer, and various value of learning rate. There were 200

training data used in this research for training the artificial neural network. The training data

were created by running the robot manually and measuring the position of each link directly.

Table I shows the experimental result in various learning rate and number of neuron per hidden

layer for the artificial neural network using 1 hidden layer. The best RMSE value is 0.01474.

Overall, the best performance is resulted by artificial neural network with 1 hidden layer and 38

neurons per hidden layer.

Table 3.1: RMSE Value of Network

No of

layers

0.1 0.3 0.5 0.7 0.9

1 0.05005 0.04715 0.04494 0.04358 0.04274

12 0.04382 0.0409 0.03936 0.0390 0.03823

25 0.04128 0.0860 0.03591 0.03535 0.03678

38 0.01474 0.03484 0.03642 0.03946 0.04029

50 0.0324 0.03658 0.04197 0.05223 0.05903



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CHAPTER 4

ADVANTAGES AND DISADVANTAGES

4.1 Advantages

1. A neural network can perform tasks that a linear program cannot.

2. When an element of the neural network fails, it can continue without any problem

by their parallel nature.

3. A neural network learns and does not need to be reprogrammed.

4. It can be implemented in any application.

5. It can be implemented without any problem.

4.2 Disadvantages

1. The neural network needs training to operate.

2. The architecture of a neural network is different from the architecture of

microprocessors therefore needs to be emulated. Requires high processing time

for large neural networks.



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CHAPTER 5

CONCLUSIONS

From the experimental result, it can be concluded that the artificial neural network can be used to

model the inverse kinematic of robot manipulator. By using artificial neural network, we do not

need to derive mathematical model of inverse kinematics of robot. The inverse kinematic of

robot is modeled into form of number of hidden layer, number of neurons per layer, weight and

bias value of the network. In this case, the best model is achieved by artificial neural network

with 1 hidden layer, 38 neurons per layer. And this model gives RMSE value of 0.01474.

Thus when the traditional method for erecting a robot manipulator becomes extremely difficult,

we need to go for the ANN technology where we dont need to explicitly write programs for the

robot. Instead, the robot learns through a trial and error method. This saves a lot of programming

time as well as the cost of constructing a robot.



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REFERENCES

Thiang, Handry Khoswanto, Rendy Pangaldus,Artificial Neural Network with Steepest Descent

Back propagation Training Algorithm for Modeling Inverse Kinematics of Manipulator, WorldAcademy of Science, Engineering and Technology 60 2009

DEPARTMENT OF BIOTECHNOLOGY SCOE PUNE

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