Seminar 3

1. INTRODUCTION

The Inverted Pendulum is one of the most important classical problems of Control

Engineering. It is a well known example of a nonlinear and unstable control problem.

Inverted pendulum being an inherently unstable system is often used as a benchmark for

verifying the performance and effectiveness of a control algorithm [6].

These systems are of two types cart-pendulum and rotary inverted pendulum. It is

virtually impossible to balance a pendulum in the inverted position without applying

some external force to the system. In cart inverted pendulum this force is applied to cart

while in rotary IP to the horizontal link. The rotary inverted pendulum is a simple

structure, multi-variable and unstable non minimum phase system subjected to many

nonlinear characteristics. It has a pendulum attached to a rotary arm instead of a moving

cart. The advantage of this system is that there is no end point, which makes it convenient

for experimentation especially during velocity control of the arm speed [8].

The Inverted Pendulum problem resembles the control systems that exist in robotic arms.

The dynamics of Inverted Pendulum simulates the dynamics of robotic arm in the

condition when the center of pressure lies below the centre of gravity for the arm so that

the system is also unstable. The common thread among these systems is to balance a link

on end using feedback control [10]. Inverted Pendulum is an abstract model for many

existing control systems such as space booster rocket or satellite or missiles guidance

systems, an automatic aircraft landing system, aircraft stabilization in the turbulent air-

flow, stabilization of a cabin in a ship, heavy Cranes lifting containers in shipyards[7], self

balancing Robots, for future transport systems like segways and jetpacks.

SCoE, Mechanical (Mechatronics)1

2. LITERATURE SERVEY

Yi-Wei Tu, Ming-Tzu Ho have designed and implemented robust real-time visual

servoing control with an FPGA-based image processor for a rotary inverted pendulum.

The position of the pendulum is measured with a machine vision system. The pendulum

used in this study is much shorter than those used in published vision-based pendulum

control system studies, which makes the system more difficult to control. The design of

the stabilizing controller is formulated as a problem of the mixed H2/Hinf control, which

is then solved using the linear matrix inequality (LMI) approach. The designed control

law is implemented on a digital signal processor (DSP) [1].

Author Faiza Faizan, Faizan Farid, Muhammad Rehan, Shoaib Mughal, M Tahir Qadri

have implemented a discrete PID controller on cart inverted pendulum. The system is

implemented using PIC microcontroller. The front panel in this system is built in

LABVIEW. The hardware model consists of sensor (incremental encoder) to sense the

position of pendulum, motor to drive the cart and circuit boards of discrete PID. The

system communicates with PC through serial communication. The system starts working

when the sensor detects the set point from the Index signal of incremental encoder. Once

PID starts then it generates commands for driving the Inverted Pendulum System on

behalf of the angle of pendulum. Pendulum is attached with the cart and the movement of

the cart is controlled by full bridge inverter. The stabilization process is completed in 10

milliseconds. The system is limited in 10 degree span i.e. it can only destabilized only if

it is under its limit (5˚). The simulation of PID is done on Matlab and simulink [2].

Zhongmin Wang, YangQuan Chen, Ning Fang implemented a minimum time swing up

control of a rotary inverted pendulum. A PID controller plus an impulse controller is

proposed in this paper for the swing up control.To make the overall control strategy more

robust, a mode switching control method is also proposed. The furuta pendulum used in

this study consists of a rotary servo motor system which drives an independent output

gear. The rotary pendulum arm is mounted to the output gear and the pendulum is


attached to the hinge. The overall controller is divided into three parts the swing up

controller, the mode switching controller and the balancing controller/regulator. For

swing up control a positive feedback PID controller is proposed. When the pendulum is

almost upright, a state fed back controller is implemented to maintain it upright and reject

the possible external disturbance using linear quadratic regulator based on the linearized

plant model. In this study the swing up time has been reduced approximately from being

longer than 8 sec. to being less than 3 sec. [3].

Mario E. Maga˜na and Frank Holzapfel has presented an experimental setup of a fuzzy-

logic controller of an inverted pendulum that uses vision feedback. The fuzzy-logic

controller used to control the experimental inverted pendulum uses conventional

triangular membership functions to fuzzify the data measured by the vision system. The

setup consists of a mechanical system composed of an inverted pendulum, a video

camera and a vision computer that are used as a non touching sensor , a fuzzy-logic

controller that is implemented on a 386 personal computer using Borland C++ and an

actuator that consists of an armature-controlled dc servo-motor driven by a pulse width-

modulated amplifier. As the camera sends the data of each frame for 15 ms and pauses

for 1.25 ms during the vertical blank, this window is used to copy the information in to

the main memory. This vertical blank period gives opportunity to manipulate the data and

compute the angle and position of the sled simultaneously before the next image is

acquired. A lead screw is directly coupled to the shaft of a dc motor and is guided by two

steel bars using bearings drives the sled [4].

3. INVERTED PENDULUM TYPES

Inverted Pendulum systems are of two types

Cart Inverted Pendulum (CIP)

Rotary Inverted Pendulum (RIP)


3.1 Cart Inverted Pendulum (CIP)

These systems are also called as Carriage Balanced Inverted Pendulum (CBIP). This

system involves a cart, able to move backwards and forwards, and a pendulum, hinged to

the cart at the bottom of its length such that the pendulum can move in the same plane as

the cart. That is, the pendulum mounted on the cart is free to fall along the cart's axis of

motion.

Figure 1. Cart-pendulum [9].

3.2 Rotary Inverted Pendulum (RIP)

Rotary inverted pendulum systems are also known as furuta pendulum. The RIP is a

simple structure, multi-variable and unstable non minimum phase system subjected to

many nonlinear characteristics. It has a pendulum attached to a rotary arm instead of a

moving cart. The advanced of this system is that there is no end point, which makes it

convenient for experimentation especially during velocity control of the arm speed. It

consists of two links, a motor driven horizontal link and an unactuated vertical pendulum

link. A DC power supply together with a pulse-width-modulated (PWM) servo amplifier

supplies power to motor.


Figure 2: Furuta or rotary inverted pendulum [9].

4. INVERTED PENDULUM CONTROL SYSTEM

The inverted pendulum control system is shown in fig. 3. The pendulum may be a cart

mounted or arm driven rotary inverted pendulum. Pendulum angle measurement is done

by sensors like camera, encoder or potentiometer. The angle feedback block gives actual

pendulum angle which is compared with reference angle to produce error signal or

deviation. Reference angle is kept zero or near zero as pendulum has to be balanced in

upright position. Error signal is fed to the controller block. Various controller

implementations are PID, LQR,FLC, H2/H∞. The controller output is control signal

which is motor control voltage fed to the motor driver.

Figure 3: General block diagram


Controller (PID, LQR,

FLC,H2/HINF)

Inverted Pendulum

system(CIP, RIP)

Angle Feedback(Vision,

Encoder, Potentiomet

er)

Error Control Signal

Actual Angle

Reference Angle +

-

5. CONTROL METHODS

Various methods are implemented in the literature for inverted pendulum control. Some

control methods are PID, LQR, H2/H∞, FLC. The sensors used for measurement of

pendulum angle are image sensor, encoder, potentiometer.

5.1 Using vision feedback

Inverted pendulum control systems with vision feedback use image sensor to measure

pendulum angle. This is non-contact type sensor. Other non-contact type sensor involves

radars, ultrasonic sensors, infra red sensors. Vision sensors are able to identify and track

targets within the scene with ease and provide more information about the relationship

between targets and environment [1]. Measurements made from images are noisy in

nature due to modeling uncertainty of the image sensor and changes in the environment,

background, and illumination, which deteriorates the performance of visual servoing

systems. To mitigate noise, a filter is commonly used to smooth noisy data before it is fed

to the controller.

5.1.1 Just-In-Time Control

Authors Ken-ichiro Fukuda, Shun Ushida and Koichiro Deguchi in [5] have proposed a

control system for stabilization of inverted pendulum which uses a normal frame rate

camera. The motivation behind the study is that, humans can keep a stick on the fingertip

in upright position for a certain time after a brief exercise, although humans have a large

time delay due to signal transfer in the optic nerve and due to information processing in

the brain. Correspondingly in an experiment on the stabilization of the mechanical

inverted pendulum, a normal frame rate camera is used as an angle sensor of the

pendulum, and its large time delay makes the stabilization difficult. JIT is memory-based

scheme which corresponds to the mechanism of human learning and memory.


5.1.1.1 Stabilization By PD Controller

A two-link nonlinear Direct Drive Arm and a pendulum placed on the tip of the DD-arm

is considered. To measure the pendulum angle, one camera is placed at the ceiling of

experimental laboratory.(X,Y,Z)and(u,v)indicate 3D world coordinates system and 2D

image coordinates system, respectively. Figure 4 shows the world and image coordinate.

The length of the pendulum ‘h’ is known. By assuming that the pendulum angle is very

little and the height of the pendulum head is always h in 3D world coordinates system.

The relationship between the 3D position (Xt,Yt,h) and 2D position (ut,vt).

w (ut

v t

1 )=P(X t

Y t

h1

) (1)

P is projection matrix. By taking 3D positions at the top (Xt,Yt,h) and bottom (Xb,Yb,h)

of pendulum, the angle of pendulum is calculated as

Φ≈ sin Φ=√(Xb−X t)

2+(Y b−Y t)2

h (2)


Figure 4 : World coordinates and camera coordinates [5].

5.1.1.2 Stabilization By JIT Method

In JIT modeling method prior knowledge about the complicated controlled object is not

required. In traditional online identification, the system global model is updated for every

new data. In general it is called Eager Learning. Instead of global model, JIT method

stores much past input-output data of the system. Only when it receives requests to

process of system information, it combines stored data to reply to information requests.

This identification is also called Lazy Learning. In global modeling, prior knowledge

about constructions and parameters of identified objects is considerably important and

necessity. As an another advantage, JIT method has much past input-output data of the

system. This makes one think of human memory and human determination mechanism.

In fact humans memorize their past experiences and make behavioral decisions using past

experiences. JIT method also memorizes past input-output data and predicts system

information combining stored data [5].

5.1.2 Mixed H2/H∞ control

Yi-Wei Tu, Ming-Tzu Ho have presented design and implementation of real time visual

servoing control with an FPGA-based image co-processor for a rotary inverted pendulum.

The pendulum used in the proposed system is much shorter than used in usual systems

i.e. 0.156 m which makes the system more difficult to control. The designed control law

is implemented on a digital signal processor (DSP) [1].

5.1.2.1 Machine vision system

The image processing algorithms of the machine vision system are pipelined and

implemented on a field programmable gate array (FPGA) device to meet real-time

constraints. In this paper, the position of the pendulum is measured with a machine vision

system that uses an image sensor with a frame rate of 250 frames per second. The camera


uses a KODAK CMOS image sensor, KAC-9630, which can capture 128 ×101 pixels

quantized to 256 gray-levels at 580 frames per second. The camera is mounted on the

rotating arm of the inverted pendulum system. The SRAM (2 M bits) acts as a frame

buffer, which is used for storing an image during camera calibration. The FPGA board is

based on an Altera Cyclone EP1C12Q240C8 FPGA [1]. A circular-shaped marker is

placed on the top of the pendulum to simplify visual sensing and to increase the accuracy

of visual measurements. Determination of the position of the marker is based on the

perspective pinhole camera model which is given by

λ p=K ICP [ RECP ⃒ t ] POPCS (3)

p represents two dimensional homogeneous coordinates of image point in image

coordinate system, Popcs represents the three-dimensional homogeneous coordinates of

the object point in the world coordinate system, and λ is a scalar factor. The matrix Recp

and three-dimensional vector t are extrinsic camera parameters that describe the rotation

and translation between the world frame and the camera frame, respectively. KICP is the

intrinsic camera parameter matrix.

5.1.2.2 Edge detection

Edges are the pixels at or around which there is a large change in gray-level intensity. In

most images, edges characterize object boundaries and are therefore useful for the

Segmentation and identification of objects in a scene. Most edge detection techniques are

based on the idea of computing image gradients. Let f(x, y) denote the grayscale level of

the pixel at point (x, y) in the image coordinate system. The gradient of an image

grayscale level at pixel (x, y) is defined as

∇ f ( x , y )≣

∂ f∂ x∂ f∂ y

=[ f x (x , y)f y( x , y )] (4)


In this study the computation of the image gradient is based on the Sobel operator as

shown in fig 5. The magnitude of image gradient is given by following equation,

|∇ f|≜√ f x2+f y

2 (5)

To lower computational complexity, the magnitude of the gradient is approximated using

the sum of the absolute values of fx and fy, which is given by

|∇ f|≈|f x|+|f y| (6)

Figure 5: Edge detection using sobel masks [1].

5.1.2.3 Thresholding

Once the edge strength is computed using edge detection, the next step is to apply a

threshold to decide whether edges are present at a specific point of the image. The

thresholding with a threshold value 150 is applied which is determined experimentally as

shown in fig 2. The thresholding operation separates the edge of the marker from the

background and the pendulum rod [1].


Figure 6: Threshold applied to the image [1].

5.1.2.4 Centroid and displacement determination

As the marker is symmetrical, the centroid of the edge is taken as its location.The

centroid (xc, yc) is given by,

xc=∑

x∑

y

x . f T (x , y)

∑x∑

y

f T (x , y)

(7)

where fT(x, y) represents the binary level of the pixel at point (x, y) in the image

coordinate system.


Figure 7: Angular displacement of the pendulum rod [1]

Using triangulation, the angular displacement of the pendulum rod is given by

θ=2sin−1 √( Xc−Xe )2+(Y c−Y e )

2

2 L

(8)

where L is the distance between the pivot joint and the center of the marker.

5.1.2.5 Stabilization using mixed H2/H∞ control synthesis

The design of the stabilizing controller is formulated as a problem of the mixed H2/H∞

control, which is then solved using the linear matrix inequality (LMI) approach. The

rotary inverted pendulum system has two equilibrium points. The downward pendulum is

a stable equilibrium point. The upward pendulum is an unstable equilibrium point. In

order to design the stabilizing controller, the Jacobian linearized model is needed. The

synthesis of the controller is formulated as a problem of the mixed H2/H∞ control. The

H∞ norm constraint is used to enhance robustness against dynamic uncertainty, and the

H2 performance is used to ensure control effort, disturbance attenuation, and sensor noise

attenuation. The simulations are carried out in MATLAB/Simulink. To examine the

robustness of the proposed control system, an impulsive disturbance was manually added

by tapping the pendulum at a time of 3.5 s, the control system recovers from the

impulsive disturbance. The pendulum slightly oscillates in the upright position within a

range of ±1º, and the rotating arm oscillates within a range of ±6º [1].

5.2 Using Encoder feedback


The pendulum angle is measured using incremental encoder and angular velocity of

pendulum is obtained by taking its derivative. These values are fed back as system to

calculate error voltage.

5.2.1 LQR and energy based PD control

Viroch Sukontanakarn and Manukid Parnichkun have developed a rotary inverted

pendulum the mechanical model of which is derived by using Euler-Lagrange. An

Energy based on PD controller was applied in self-erecting of the pendulum while LQR

controller was applied to balance the pendulum [8]. The rotary inverted pendulum system

consists of a controller, an arm, a pendulum, an actuator (a dc motor) and two increment

rotary encoders. The controller makes the pendulum stand at upright position on the

rotary arm by moving the arm supported on the base. The motor provides power to rotate

the arm. The encoders detect the pendulum and arm angular position. Derivation of

mathematical equation describing dynamics of the rotary inverted pendulum system is

based on Euler-Lagrange equation of motion:

ddt ( ∂ L

∂ q ' i)− ∂ L

∂ qi

+∂W∂ q i

=Qi (9)

Where q(t) is angular position vector, q˙(t) is angular velocity vector, Qi external force, L

is Lagrangian and W is loss of energy. In the Euler-Lagrange equation, the Lagrangian,

L, is defined as follows

L (q , q ' )=T total−V total (10)

Where, Ttotal is Total kinetic energy of the rotary inverted pendulum system and Vtotal

is the Total potential energy of the rotary inverted pendulum system.

5.2.1.1 Self-erecting pendulum by energy based on PD controller


The self-erecting controller is used to swing the inverted pendulum up to the upright

position when the pendulum is far from the upright position; for example, when the

pendulum falls down by gravity. The self-erecting controller brings the pendulum upright

close to the unstable point of equilibrium. Swinging up controller calculates the total

system energy based on the kinetic energy of both links and the potential energy of the

pendulum. The control law function for input of the system is given by,

uswing=V a|Eref−E tot|θ ' 2cosθ2 (11)

Etot=12

J1 θ'1

2+ 12

m2 L12 θ'

12+ 1

2(m2l2

2+J 2 )θ'22+m2 L1 θ '1 l2θ '2 cosθ2+(l2θ '2 sin θ2)

2−m2 g l2cos θ2

(12)

Eref =g m2 l2 (13)

Where, Etot = Total energy in stable position of motion pendulum,

Eref = Total energy in upright stable equilibrium position.

5.2.1.2 Balancing controller design

During the self-electing of the pendulum, a robust state controller may be appropriate to

catch the pendulum at the upright position [8]. To switch between swinging up and

balancing algorithm, the normalized energy of the pendulum Etot is to be calculated and

compared with the design energy of the pendulum Eref. The Linear Quadratic Regulation

(LQR) is used for the calculation of the optimal gain matrix K such that the state

feedback law is followed. The minimized the cost function is shown in equ. 15.

ubalancing=−Kx (14)


J=12∫0

∞

(XT Qx+uT Ru)dt (15)

Using MATLAB, the LQR balancing controller for the rotary inverted pendulum is

designed based on the required weighting matrices. There are two PCs: Host and target

PCs. The host PC with MS-Windows XP OS runs matlab, simulink, real-time workshop

(RTW), xPC target and C/C++ compiler. The simulink is used to model dynamic of the

physical system and controllers. RTW and C/C++ compiler convert simulink blocks into

C code and build a target that is then download to the target PC through TCP/IP

connection. Thus the controller regulate the arm about zero degree within 120° and

balance the pendulum about the upright position within 3°. The arm is rotated alternately

between 0 and 50° to increase the energy of the system quickly to move the arm to its

maximum allowed angle of circular of arm. In this self-erecting mode, energy based PD

controller takes 4.5 sec for 5 swings [8].

5.2.2 Fuzzy logic controller

Singh Vivekkumar Radhamohan, Mona Subramaniam A, Dr. M.J.Nigam have designed a

real time control for swing-up stabilization of inverted pendulum using Fuzzy logic. In

this model a single rulebase is used to control both position and angle simultaneously

during both swing-up and stabilization. The use of fuzzy logic does not require extensive

knowledge of the system. The design of this rule base uses 3x3 rules or in other words 9

rule rulebase. The position error and angular error are combined to obtain a single error

signal ‘e’, similarly the differential position error and differential angular error are

combined to obtain a single differential error signal ‘de’. These signals are given to the

fuzzy controller thereby eliminating two different fuzzy controllers being used for the

purpose. With this the number of tuning parameters is reduced as well. Switches are to

change the gains for swingup and stabilization, just by changing the gains the mode are

changed between swing up and stabilization. During swing up mode energy is induced


into the system by triggering only those rules which increase the energy into the inverted

pendulum. This process is continued until the pendulum reaches 30º from the final

vertically upright position, the stabilizer mode of control comes into effect at this point

[6].

5.2.2.1 Swing-up routine

In the method proposed in [6] the aim is to swing up the pendulum from vertical

downward position to a ± 30º of the vertical position. While swinging up care is taken

that the pendulum does not cross the rail limits as it is a limited rail length situation. The

basic strategy is to move the cart in such a way that energy is slowly pumped into the

pendulum. This is achieved by satisfying a particular mathematical condition derived

from the mechanical energy equations while designing the rule base of fuzzy control

system. The total mechanical energy of the pendulum and its derivative E’ are given by

equation

E=12

m L2θ ' 2+mg L(1−cosθ) (16)

E'=ml θ' (cosθ) x ' ' (17)

From the above equation it follows that energy E can be pumped or removed from the

system by changing the sign of E’. Now E’ is positive if X’’ > 0 and θ’cosθ > 0. Similarly

E’ is negative if X’’ < 0 and θcosθ > 0 or X’’> 0 and θ’cosθ < 0. So to pump the energy

into the system we design our rulebase such that E’ is positive most of the times and

never negative. Energy is pumped into the pendulum slowly and the pendulum is swung

up to nearly vertically upright position.


5.2.2.2 Stabilization

Once the pendulum is swung within ± 30˚ of the vertical position the fuzzy stabilization

controller takes over from swing up controller. The proposed fuzzy stabilization

controller uses two input variable e and e’. Here is the differenceof cart velocity X’ from

the pendulum angular velocity θ’. Instead of using a separate rule base for stabilization

the gains are switched to another value, while the same fuzzy rule base is used [6].

6. APPLICATIONS

Inverted pendulum is used as abstract model in many applications such as

Attitude control of a space booster rocket or satellite or missiles guidance where

thrust is actuated at the bottom of a tall vehicle.

Future transport systems like Segway. The Segway is a two-wheeled, self-

balancing electric vehicle.

Self balancing robots like humanoid robots.

Crane lifter widely used in industrial plants and rehabilitation equipment [7].

Central component in the design of early seismometer. Seismometers are

instruments that measure motions of ground, including those of seismic waves

generated by earthquake, volcanic eruptions other seismic sources.


7. SUMMARY

Inverted Pendulum is an example of a nonlinear and unstable control problem. In real

time control problem, first the pendulum is swung up or manually positioned upright that

is in a position of unstable equilibrium. The controller is then switched in to balance the

pendulum and to maintain this balance in the presence of disturbances. A simple

disturbance may be a light tap on the balanced pendulum. The pendulum will be balanced

by using feedback angle. Control with vision feedback resulted in very slight oscillations

that is ±1º using higher FPS camera [1] while using fuzzy logic, pendulum was kept

upright for short amount of time due to lower frame rate [4].Using JIT approach [5]

realization human memory and determination mechanism in motor control is done which

is designed for both time delay compensator and stabilization. With encoder feedback

using energy based PD controller it takes 4.5 sec. for 5 swings which maximum swing in

short time, there were still some small amplitude oscillations around equilibrium point.


REFERENCES

[1] Yi-Wei Tu, Ming-Tzu Ho, Design and implementation of robust visual servoing

control of an inverted pendulum with an FPGA-based image co-processor,

Mechatronics, 2011, Vol.10, No. 3 , pp. 01-13.

[2] Faiza Faizan, Faizan Farid, Muhammad Rehan, Shoaib Mughal, M Tahir Qadri,

Implementation of Discrete PID on Inverted Pendulum, 2nd International

Conference on Education Technology and Computer (ICETC), Karachi, Pakistan,

2010, pp. 48-51.

[3] Zhongmin Wang, YangQuan Chen, Ning Fang, Minimum-Time Swing-up of A

Rotary Inverted Pendulum by, Iterative Impulsive Control, Proc. of the 2004

American Control Conference, Boston, Massachusens, 2004, pp. 1335-1340.

[4] Mario E. Maga˜na and Frank Holzapfel, Fuzzy-Logic Control of an Inverted

Pendulum with Vision Feedback, IEEE transactions on education, 1998, Vol.41,

No.2 , pp. 165-170.

[5] Ken-ichiro Fukuda, Shun Ushida and Koichiro Deguchi, Just-In-Time Control of

Image-Based Inverted Pendulum Systems with a Time-Delay, SICE-ICASE

International Joint Conference Bexco, Busan, Korea, 2006, pp. 4016-4021.

[6] Singh Vivekkumar Radhamohan, Mona Subramaniam A, Dr. M.J.Nigam, Fuzzy


Swing-Up and Stabilization of Real Inverted Pendulum Using Single Rulebase,

Journal of Theoretical and Applied Information Technology, 2010, Vol.3, No.2,

pp.43-50.

[7] Takami Matsuo, Ryoichi Yoshino, Haruo Suemitsu, and Kazushi Nakano,

Nominal Performance Recovery by PID+Q Controller and Its Application to

Antisway Control of Crane Lifter With Visual Feedback, IEEE Transactions On

Control Systems Technology, 2004, Vol.12, No.1, pp. 156-166.

[8] Viroch Sukontanakarn and Manukid Parnichkun, Real-Time Optimal Control for

Rotary Inverted Pendulum, American Journal of Applied Sciences, 2009, Vol.6,

No.6, pp.1106-1115.

[9] Mun-Soo Park and Dongkyoung Chwa, Swing-Up and Stabilization Control of

Inverted-Pendulum Systems via Coupled Sliding-Mode Control Method, IEEE

Transactions On Industrial Electronics, 2009, Vol.56, No.9, pp. 3541-3555.

[10] Jyoti Krishen, Victor M. Becerra, Efficient Fuzzy Control of a Rotary Inverted

Pendulum Based on LQR Mapping, Proc. of the 2006 IEEE International

Symposium on Intelligent Control Munich, Germany, 2006, pp. 2701-2710.


Seminar 3

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