A transfer function is  defined as a ratio of two polynomials:

Where N(s) and D(s) are simple polynomials

These can be writtern as:

N(s)=(s − z1)(s − z2) . . . (s − zm−1)(s − zm)
D(s)=(s − p1)(s − p2) . . . (s − pn−1)(s − pn) ,

Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.

Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s.

All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must be either purely real, or appear in complex conjugate pairs.

Poles and Zeros of a transfer function are the frequencies for which the value of the transfer function becomes infinity or zero respectively.As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.

A system is characterized by its poles and zeros .Because the transfer function completely represents a system differential equation, its poles and
zeros effectively define the system response. In particular the system poles directly define the components in the homogeneous response.The locations of the poles, and the values of the real and imaginary parts of the pole helps to determine the response of the system.

The stability of a linear system may be determined directly from its transfer function. An nth order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases.In order for a linear system to be stable, all of its poles must have negative real parts.

Reference:

http://en.wikibooks.org/wiki/Control_Systems/Poles_and_Zeros

]]> http://athulmv.wordpress.com/2009/10/01/effect-of-adding-a-zero-to-a-system-2/feed/ 0 athulmv untitled untitled http://athulmv.wordpress.com/2009/10/01/answer-2/ http://athulmv.wordpress.com/2009/10/01/answer-2/#comments Thu, 01 Oct 2009 08:05:39 +0000 athulmv http://athulmv.wordpress.com/?p=30 ]]> Optical encoders are frequently used in control systems to convert linear or rotary displacement into digital code or pulse signals.These are usually used as feedback components in the control system.Encoders are two types:

Absolute encoders and incremental encoders

Incremental encoders

Incremental encoder produce an output which is a pulse for each increment of resolution but these make no distinction between increments.An incremental encoder typically has four parts:

A light source(LED)

A rotary(or translator )disc

A stationary mask

A sensor (photodiode)

The disc has alternate opaque and transparent sectors of equal width which is etched by means of a photographic process on to a plastic disc(slots are cut out if it is a metal disc).As the disc rotates during half of the increment cycle the transparent sectors of rotating and stationary discs come in alignment permitting the light from the LED to reach  the sensor and thereby generating an electrical pulse.For fine resolution encoders ,multi-slit mask is often used to maximize the reception of shutter light.

The waveform of the sensor output of an encoder is generally triangular or sinusoidal depending upon the resolution required.Square wave signal compatible with digital logic are obtained from it by means of linear OPAMP and comparator.Alternate transparent/opaque sectors of the disc and the square wave pulse form (obtained after signal processing) in synchronous with the disc is shown in figure.The resolution of such an incremental encoder is given as:

Basic resolution=360/N

N:number of sectors of disc;each sector is half transparent and half opaque.

In a dual channel encoder two optoelectronic channels are employed.These are installed in the same rotating disc and the mask but displaced at 90⁰ to each other such that the two pulse output signals have a relative times phase displacement of 90⁰electrical.A circuit that senses the relative time phase of the outputs of the two channels determines the direction of rotation of the disc or the encoder shaft.

The output of the encoder is fed to a counter which counts the number of pulses;the count being the measure of angle(or translation)through which the encoder shaft has rotated.By sampling the counter at regular intervals by means of clock pulses it is possible to compute the speed of encoder shaft.

The function of the error detector is to give out signal proportional to the difference of input and output.The most commonly usederror detectors in servo systems are potentiometers and a pair of synchros.

A synchro is an electromagnetic transducer commonly used to convert angular position of shaft into an electrical signal.It is commercially known as a selsyn or an autosyn.It basically consists of a synchro transmitter (generator) and a synchro receiver(control transformer).

Synchro generator

Its construction is similar to that of a three phase alternator.

An ac voltage V_r(t)=V_rsinw_ct

is applied to the rotor winding through slip rings.

This voltage causes a flow of magnetising current in the rotor coil which produces a sinusoidally time varying flux directed along its axis and distributed nearly sinusoidally in the air gap along the stator periphery.Because of the transformer action,voltages are induced in each of the stator coils.As the air gap flux is sinusoidally distributed,the flux linking any stator coil is proportional to the cosine of the angle between rotor and stator coil axes and so is the voltage induced in each stator coil.Thus the synchro transmitter acts as a single phase transformer in which the rotor coil is the primary and the stator coil forms three secondaries.

Let V_s1n,V_s2n,V_s3n be the voltages induced in the stator coils S₁,S₂,S₃ with respect to neutral.When the rotor axis make an angle theta with the axes of the stator coils S₂.

V_s1n=KV_rsinw_ctcos(q+120)

V_s2n=KV_rsinw_ctcos(q)

V_s3n=KV_rsinw_ctcos(q+240)

The three terminal voltages of stator are:

V_s1s2=1.73KV_r sin(q+240)sinw_ct

V_s2s3=1.73KV_r sin(q+120)sinw_ct

V_s3s1=1.73KV_r sin(q)sinw_ct

When theta(q)=0,it is seen that maximum voltage is induced in the stator coil s2 and terminal voltage Vs3s1 is zero.This position of the rotor is defined as the electrical zero of the transmitter and is used as reference for specifying the angular position of the rotor.

The output of the synchro transmitter is applied to the stator of a synchro control transformer.Circulating currents of the same phase but of different magnitudes flow through the two sets of stator coils.The control transformer flux axis being in the same position as that of the synchro transmitter rotor,the voltage induced in the C.T rotor is proportional to the cosine of the angle between the two rotors:

e(t)=KVrcosØ sinw_ct

Ø=angular displacement between the two rotors(error angle).

when Ø=90 ,e(t)=0,ie voltage induced in the C T rotor is zero.This is electrical zero of the C.T rotor.

The synchro transmitter-control transformer pair acts as an error detector giving a voltage signal at the rotor terminals of the control transformer proportional to the angular difference between the transmitter and control transformer shaft positions.

Synchro and stepper motor

There are two distinctly different ways of using stepper motors in control systems.One is the open loop mode and other is the closed loop mode.

The stepper motor is a digital  device whose output in shaft angular displacement is completely determined by the number of input pulses.Consequently,there is no need for a feedback device to determine the position of motor  shaft and ,therefore,of the load connected to the motor shaft.We can use an open step servo system with the same accuracy as that of a closed loop analog system.

If  we need to operate the stepper motor in closed loop(positional feedback)mode,we need to use synchros for error detection.Here the motor is used like conventional servomotor.A signal from the output is fed back and is used to operate a gate controlling the pulses from a pulse generator.This is shown in the figure below:

CINCINNATI MILACRON T3 ROBOT ARM

INTRODUCTION

Cincinnati Milacron built large industrial robots primarily for welding industry. It was one of the first companies to change from hydraulic to electric robots. Milacron pioneered the first computerized numerical control (CNC)  robot with improved wrists and the tool centre point (TCP) concepts. The first hydraulic machine, the T3 , was introduced in 1978. It closely resembled the General Electric Man-mate, ITT arm, and other predecessors (Sullivan 1971). Constructed of cast aluminium, it is available in two models of 6-axes revolute jointed arms. The largest, the T3-776, uses ballscrew electric drives to power the shoulder and elbow pitch. The ballscrews replaced the hydraulic cylinders originally used on the T3 robots. The elbow is a classical example of intermediate drive elbow. The same techniques, only upside down, appear in the shoulder. Shoulder yaw is provided by the standard bullgear on a base mounted motor drive. End users have discovered that ballscrews are not sufficiently reliable and are pressuring for an alternators. The eventual disappearance of ballscrews in industrial robots seems inevitable.

This robot is a more classically designed industrial robot.   Designed as a healthy compromise between dexterity and strength this robot was one of the ground breakers, in terms of success, in factory environments.  However, while this robot was a success in industry its inflexible interfacing system makes it difficult to use in research.

CONTROL SYSTEM

The T3 robotic arms is controlled  using a Hierarchical Control System.A Hierarchical control system is partitioned vertically into levels of control. The basic comand and control structure is a tree, configured such that each computational module has a single superior, and one or more subordinate modules. The top module is where the highest level decisions are made and the longest planning horizon exists. Goals and plans generated at this highest level are transmitted as commands to the next lower level where they are decomposed into sequences of subgoals. These subgoals are in turn transmitted to the next lower control decision level as sequences of  less complex but more frequent commands. In general,the decisions and corresponding decompositions at each level take into account: (a) conrmands from the level above, (b) processed sensory feedback information appropriate to that control decision level, and (c) status reports from decision control modules at the next lower control level.

The hierarchical control structure serves as an overall guideline for the architecture and partitioning of a sensory interactive robot control system.

 CMI T3-776 CONTROL SYSTEM BLOCK DIAGRAM

The figure shown above depicts  the schematic block diagram of the integrated control structure as configured on the Cincinnati Milacron T3 Robot. The system is configured in the hierarchical manner and includes five major subsystems:

(1) The Real-Time Control System (RCS)

(2) The commercial. T3 Robot equipment

( 3 ) the End-Effector System

(4) The Vision System

(5) The Watchdog Safety System

The Real-Time Control System as shown in figure is composed of four levels:

(1) The Task Level

(2)The Elemental-Move Level

(3) The Primitive Level

(4)The T3 Level.

The Task, Elemental-Move and Primitive levels of the controller are considered to be Generic Control Levels. That is, these levels would remain essentially the same regardless of the particular robot (commercial or otherwise) being used. The T3 Level, however ,uses information and parameters particular to the T3 Robot and is, therefore, unique to the T3 Robot. The Joystick shown provides an alternate source of commands to the Primitive Level for manual control of the robot and is not used in conjunction with the higher control levels .The T3 Controller shown in figure  is part of the T3 Robot equipment as purchased from Cincinnati Milacron. This controller is subordinate to the T3 Level of the RCS and communicates through a special interface.

The End-Effector System consists of a two fingered gripper equipped with position and force sensing .The gripper is pneumatically actuated and servo controlled by a controller which is subordinate to the Primitive Level of the RCS. There are three sensory systems on the robot:

(1)The finger force and position sensors on the gripper which report data to the End Effector Controller

(2)The 3 point Angle Acquisition System which reports data to the T3 Controller, the T3 Level of the RCS and to the Watchdog Safety System

(3)The Vision System which reports data to the Elemental-Move Level of the RCS.

Of the sensor systems, the vision system is obviously the most  complex. It performs sophisticated image processing which requires substantial computational time.

The Watchdog Safety System does not fit directly into the hierarchical control structure. It is an independent system which monitors robot motions and compares them to previously defined limits in position, velocity and acceleration. The Watchdog System has the power to stop the robot if any limits are exceeded and consequently monitors both the mechanical and control systems of the robot.

PARTS OF THE REAL TIME  CONTROL SYSTEM

(1)Task Level

The Task Level interfaces with the Workstation Level above it and the Elemental-Move Level below it. In the current configuration, the Task Level has no direct interfaces with sensory systems. The Task Level receives commands from the Workstation Level in terms of objects to be handled and named places in the workstation.

For example, the task might be to find a certain part on the tray at the load/unload station, pick it up and put it in the fixture on the machine tool. This task could be issued as one command from the Workstation Level to the Task Level of the RCS.

(2)Elemental-Move Level

The E-Move Level interfaces with the Task Level above it and the Primitive Level below it. In addition, the E-Move Level interfaces with the Vision System from which it acquires part position and orientation data. The E-Move Level receives commands from the Task Level which are elemental segments of the Task Level command under execution. These are generally single moves from one named location to another. If a part acquisition is involved, data from the Vision System is requested to determine the exact location of the next goal point. The E-Move Level then develops a trajectory between the new goal point and its current position. A trajectory maybe simply a straight line move to the goal point or a more complex move, involving departure, intermediate and approach trajectories. These trajectories can be constructed using pre-stored trajectory segments or data acquired from the Vision System. If no pre-stored segments are found for the desired move and the use of vision data is not appropriate, then a straight line path to the new goal point is calculated.

(3)Primitive Level

The Primitive Level interfaces with the E-Move Level above it and the T3 Level and End-Effector Controller below it. The Primitive Level is the lowest level in the RCS which is robot or device independent. Subsystems subordinate to the Primitive Level are considered to be at the device level in the control hierarchy. In this system, these subsystems or devices are the robot and the end-effector. T3 The Level shown in figure  is not a true control decision level by itself and could be logically combined with the T3 Controller at the device level. The robot and end-effector are, therefore, at the same control decision level subordinate to the Primitive Level. Additionally, the Primitive Level interfaces with the Joystick. The Joystick is a peripheral device which is used for manual operation of the robot. Using the Joystick, the operator can control robot motion in several coordinate systems (world, tool or individual joint motions). Under Joystick control the human operator assumes the higher level planning and control duties normally handled by the E-Move and Task Levels when the robot is operating automatically. The actual Joystick unit has groups of small joysticks, rotory and rocker switches dedicated to each coordinate system. These are configured  such t hat  the  robot  will move basically the way the lever is pushed or the switch turned that the robot will move basically the way  the lever is pushed or the switch turned, giving the operator  a    relatively  feel  for  the  motion  produced ’The Primitive Level receives commands from the E-Move L e v e l  in terms of goal points in Cartesian space.These points differ  from those received by the E-Move Level from the Task Level in that they are not named locations and therefore  assume no knowledge of the Workstation layout. These points are typically more closely spaced than those at the higher Levels although this is not necessarily the case.

(4) T3 Level

The T3 Level interfaces with the Primitive Level above it and the commercial Cincinnati Milacron T3

Robot Controller below it. In addition there is a sensory interface which supplies the six individual joint angles.

The T3 Level is so named because elements of  it are peculiar to the T3 Robot. From a control hierarchy point of view the T3 Level does not constitute a logical control decision level but is infact a “gray box” necessary to transform command and feedback formats between the Primitive level and T3 controller.

References:Wikipedia,Brittanica,IEEE,Google(Pictures)

Servomechanism or servo is meant for feedback control systems where the controlled variable is mechanical position or time derivative of position. Eg: velocity, acceleration etc. It is a type of feedback arrangement for the automatic self-regulation of an electrical, mechanical, or biological system by returning part of its output as input. The constant speed control system of a DC motor is a servomechanism that monitors any variations in the motor’s speed so that it can quickly and automatically return the speed to its correct value. Servomechanisms are also used for the control systems of guided missiles, aircraft, and manufacturing machinery.

Why should we go for servo???

(1) Accurate control of motion without the need for human attendants (automatic control);
(2) Maintenance of accuracy with mechanical load variations, changes in the environment, power supply fluctuations, and aging and deterioration of components (regulation and self-calibration);
(3) Control of a high-power load from a low-power command signal (power amplification);
(4) Control of an output from a remotely located input, without the use of mechanical linkages (remote control, shaft repeater).
                                                                   

                                       A common type of servo provides position control. Servos are commonly electrical or partially electronic in nature, using an electric motor as the primary means of creating mechanical force. Other types of servos use hydraulics, pneumatics, or magnetic principles. Usually, servos operate on the principle of negative feedback, where the control input is compared to the actual position of the mechanical system as measured by some sort of transducer at the output. Any difference between the actual and wanted values (an “error signal”) is amplified and used to drive the system in the direction necessary to reduce or eliminate the error.

Block Diagram

The main aim in the above system is to keep the output as per the reference input. For this we use a feedback loop.
Here feedback signal ‘b’ (which is a function of output) is fed to the summing device which compares it with the reference input ‘r’ and produces the error signal if necessary. Error signal also known as the actuating signal is amplified first and it then actuates the servo motor which in turn controls the position of the controlled variable through gears.

Now its time 2 apply!!!

•  In Communication Satellite

                                         A typical system using a servomechanism is the communications-satellite-track antenna of a satellite Earth station. The objective is to keep the antenna aimed directly at the communication satellite in order to receive and transmit the strongest possible signal. One method used to accomplish this is to compare the signals from the satellite as received by two or more closely positioned receiving elements on the antenna. Any difference in the strengths of the signals received by these elements results in a correction signal being sent to the antenna servomotor. This continuous feedback method allows a terrestrial antenna to be aimed at a satellite 37,007 km (23,000 miles) above the Earth to an accuracy measured in hundredths of a centimeter.

•  RC SERVOS

                                  RC servos are hobbyist remote control devices servos typically employed in radio-controlled models, where they are used to provide actuation for various mechanical systems such as the steering of a car, the flaps on a plane, or the rudder of a boat.RC servos are composed of an electric motor mechanically linked to a potentiometer. Pulse width modulation (PWM) signals sent to the servo are translated into position commands by electronics inside the servo. When the servo is the commanded to rotate, the motor is powered until the potentiometer reaches the value corresponding to the commanded position.Due to their affordability, reliability, and simplicity of control by microprocessors, RC servos are often used in small scale robotics applications. The servo is controlled by three wires: ground, power and control. The servo will move based on the pulses sent over the control wire, which set the angle of the actuator arm.
                     Servomechanisms were first used in military and marine navigation equipment. Today they are used in automatic machine tools, satellite-tracking antennas, celestial-tracking systems on telescopes, automatic navigation systems, and antiaircraft-gun control systems. The design of servomechanisms is considered to be a branch of both robotics and cybernetics.
                   The position control system have innumerable application, namely, machine tool position control, constant- tension control of sheet rolls in paper mills, control of sheet metal thickness in hot rolling mills, radar tracking systems, missile guidance systems, inertial guidance, roll stabilisation of ships etc.
                    Advancement in servomechanism has led to the development of the new field of control and automation, the robots and robotology.Servo favours robotics with its high performance in a hazardous environment like radioactive area. SimpleServo can replace complex mechanical systems, such as lineshafts and cams, clutches and brakes, pneumatic and hydraulic cylinders.This increases machine speed, accuracy and flexibility, while reducing build, maintenance and changeover times.