Joint Torque Reduction of a Planar Redundant Manipulator Thesis

7/24/2019 Joint Torque Reduction of a Planar Redundant Manipulator Thesis

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The University of Nottingham

Faculty of Engineering

Department of Electrical and Electronic Engineering

JOINT TORQUE REDUTION OF ! "#!N!R REDUND!NT

$!NI"U#!TOR%

AUTHOR : SYED ARHAM ABBAS RIZVI

ID : 010554

SUPERVISOR : Dr. SAMER YAHYA

MODERATOR : Dr. LIM CHIN KEONG

DATE : APRIL, 11, 2014

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T!r" #$%r &r'($)* &r'&'+% +-/!**$" ! &%r*!% -!/$* ' *$ r$-!r$/$*+ ' *$

"$3r$$ ' &achelor of Engineering

Acknowledgement

First and foremost, I would like to thank the Almighty for giving me strength and determination so that I was

able to complete my project.

Without a doubt, I would like to thank Dr. amer !ahya, as without his knowledge and complete support I

would not have been able to accomplish the task set out to me. And his cheerful attitude every time I met him

motivated me to further my project. "hank you for all of your hard work and time dedicated on me.

"his thesis, without any further thought is dedicated to my wonderful parents who have given me full support in

all issues I have undergone in life. "hey sacrificed a lot just to let me study and have a better future. I would like

to thank my beloved sister who supported in times of need so that I would be able to complete this project.

I thank my friends for always being with me and giving me the re#uired courage in order for me to complete my

project as well as my thesis.

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Table of ContentsAcknowledgement 1

Abstract 3

Introduction 4

Objectives 5

Progress and Achievements 5

Thesis Structure

!iterature "eview #

"ypes of $anipulator %

"ypes of &oints '

$otors (sed In $anipulators )

"ypes of "ransmission ystems in $anipulators *+

onventional $anipulators **

"endon-based "ransmission ystem *

/elt-based "ransmission ystem *'

0ear-based "ransmission ystem *)

1ydraulic23neumatics-based "ransmission ystem

$ethodolog% &4

$anipulator Design and 3ro 4ngineer $odelling 5

/evel-0ear "ransmission ystem 5

"endon-3ulley "ransmission ystem 6+

alculations 6

"esults 3'

/evel-0ear $anipulator 6)

"endon $anipulator 7*

onventional $anipulator 76

omparing the designs 75

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(iscussion and Anal%sis 4)

*onclusion 5+

,urther -ork 5+

"e.erences 51

A//endi0 53

Abstract

A manipulator is an engineering tool that allows a user to manipulate their environment via it. As there are a

variety of manipulators which are used for specific task according to their design and si8e, it has come to notice

that the overall design can be improved in terms of efficiency i.e. power output to power input.

"his project is about the design of a new mechanism which would be more efficient compared to theconventional manipulator as the weight of the mobile i.e. the links would be reduced, as the stepper motor,

which are usually placed at the link would now be placed at the base, thus the weight of the motor would now be

removed which contributes the most to the link9s weight. :arious transmission systems can be used instead of

the conventional design but there are limitations in those designs as to how much tor#ue they can transmit, how

much area they cover, etc. "hus the bevel gear and tendon transmission system were modelled and compared to

the conventional design.

"he tendon based transmission had the minimum mass thus the tor#ue re#uired to move it was lowest. At the

position where the tor#ue re#uired is ma;imum, a tor#ue of *.


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Introduction

In this world, energy is a very valuable and finite resource and new ways to harness it are being researched and

developed. "he whole process of energy is being developed to be more efficient so that energy in all its forms

can be harnessed more efficiently thus more energy is available to be used. >esearch and Development

departments of every company are finding out new ways to make their devices use less power than their

previous models to perform the same task thus increasing their efficiency and thus needing less energy on theoverall thus more people would be able to use them.

"o decrease the energy used in devices, new technology is developed and new methods of to convert the energy

into useful work are researched. "he most widespread form of energy in this world is electricity as it is a key

factor in industriali8ation, urbani8ation ?creation of new 8ones for people to live a life in a healthy lifestyle@,

economic growth and increasing the #uality of lifestyle of people.

4lectricity is mostly created by combusting hydrocarbons in the form of petroleum and natural gas or using coal

as the raw resource. As these are finite resource and research shows that their #uantity is decreasing and would

eventually finish in a predicted 5+ years9 time. "he figure below shows the types of resources and how much

they contribute to the creation of power on a global scale.

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37%

21%

25%

8%

9%

Petroleum Coal Dry Natural Gas Renewable Resources Nuclear-Electrcty

,I 12+1 A global creation o. /ower according to resource t%/e in &++)2 1

Devices everywhere are getting more efficient and so are manipulators. $anipulators have been modified

countless times and in numerous different ways so that they are able to provide the same power output with less

input power. In this project, the same thing is to be done so that the manipulator modelled would have an overall

better efficiency than the manipulators already in use.

$any models of manipulators were designed which would theoretically re#uire less tor#ue to operate and thesewere compared to each other and the best design was chosen. "his design was then compared to the normal

manipulator design. "he results showed that theoretically, the design would improve the efficiency of the

manipulator. Further discussion would be done in the methodology section.

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Also as normal manipulators have the motors at the link, the mass of the overall manipulator would increment

considerably with each degree of freedom ?DF@ thus for the tor#ue re#uired to be minimum, the DF of the

manipulator are kept to the minimum so that the manipulator can perform its re#uired tasks. "his decreases the

manipulator9s workspace giving it less freedom of movement. If there are obstacles in the manipulator9s path,

then the manipulator might be hindered if the obstacle blocks the workspace. "his can be improved with

increased DFs but a conventional manipulator is not able to achieve that as efficiency is lost as the mass of the

manipulator is increased for the task which can be done by a less DF manipulator. "hus a new design of the

manipulator is needed to perform the task with increased DFs so it can have a larger volume for its workspaceso that obstacle hindrance would be minimi8ed.&

Obecti!es

"he objectives of this project are as followsB

*. >esearching the designs of the manipulators and their advantages and disadvantages over other models.

. $odelling a robot manipulator design in C3ro-4. "his was changed from the software olidworks as the

university does not have the license for the software.6. 3erform simulations on the design.

7. alculate the dynamic e#uations of a $anipulator

5. ompare the modelled design with the motors at link design.


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T"esis $tructure

"his paper is organised into a few topics which are listed below as followsB

Abstract

"his part describes the reason for creating more efficient designs.

Introduction

/riefly introduces the topic and gives a very short usage detail of the manipulator

bjectives

0ives the things to be achieved during the duration of the project

3rogress and Achievements

0ives information of what was achieved in the project.

"hesis tructure

0ives a brief description of each title in the thesis

Eiterature >eview

3rovides the information of where all the information was collected from for the designs of the

manipulators.

$ethodology

0ives the details of how the design was modelled, how the calculations were prepared and processed and

how the designs were compared.

Discussion and Analysis

4;plains the result of the comparison between the models

onclusion

0ives a small e;planation of the overall achievements of the research and what could be done on it in the

future.

>eferences

"his section gives the list of citation of the source of knowledge which were collected and combined in this

paper.

Appendi;"his section provides the mathematical formulas and the results from $athematica.

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&iterature 'e!iew

"his will detail all the information gathered to create the design which was finalised and all the other models.

T%/es o. $ani/ulators

"here are main types of manipulators which are used. "here are the serial and planar manipulators.3

3lanar manipulators have separate kinematic chains that are linked to the moving platform and the ground at the

same time. "hey have an advantage over their serial counterparts as they are able to produce more precise

movement and they are able to carry higher loads and they have a better build as they have better rigidity. "hey

can also produce higher velocity and accelerations thus being able to perform the task faster and more

effectively. erial $anipulators have simple forward kinematics while the inverse kinematics for them is #uite

difficult whereas for parallel manipulators, the forward kinematics is harder than the inverse kinematics.

,I12+& A t%/ical (O, serial /arallel

mani/ulator2 4

3lanar 3arallel $anipulators perform translations along the ; and y a;es and the rotation around the 8 a;is.

"hey have the advantage of micro-minimi8ation and pick and place operations as their kinematics operations are

easier to perform than their non-planar counterparts.&

3arallel $anipulators have a comple; closed loop chain mechanism thus the kinematic analysis of the parallel

manipulator is more difficult than the serial counterpart.

"he serial parallel manipulator shown in FI0*.+ has both the advantages of erial and 3arallel $anipulators

from rigidity to workspace. "hese type of manipulators haven9t been investigated deeply though.&

erial manipulators on the other hand are the simplest type of manipulators and have been used in almost all the

application as they are easy to design, model and create. A link model of the manipulator is shown in FI0*.+6.

"hey consist of links and joints and are connected in series to one another.

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,I12+3 The link model o. a serial 3 (O, mani/ulator25

T%/es o. 6oints

&oints are the referred to two bodies as they are connecting bodies and providing them a motion relative to one

another. "hese two bodies can be said to be frames of reference. "he *stframe is to be fi;ed while the ndframe

is to be mobile. "hey are fundamental to a robot as without them, the robot would just be a rigid structurewithout any motion.

T)*es of+oints

T)*es of,otion

-egree of.reedom 'elati!e Position

Re!olute Rotatonal 1 D"# $nular &s'lacement ()*Prsmatc $+al 1 D"# ,near &stance

crewRotatonal an&$+al 1 D"#

$nular &s'lacement an& ,near&stance

'.ercal Rotatonal 3 D"# $nular &s'lacement () an& /*

Rotatonal Rotatonal 1 D"#

$nular &s'lacement

(rotaton alon a+al a+s*"rt.oonal $+al 1 D"#

,near &stance alon a 90 anular&s'lacement rom 1strame

TA712+1 !ist o. joints and their con.iguration2

,I12+4 6oint (iagrams2 #

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$otors used in $ani/ulators

$otors are the main energy conversion devices ?electrical energy to mechanical energy@ used in

manipulators and they are the most e;pensive parts of the manipulator normally. "hey convert electrical

impulses to angular rotation. "here are a variety of motors ranging from eries D $otor to A >eluctance

motors and they are used for their respective applications. "he motor used in manipulator mainly is the

stepper motor.

As most applications re#uire precise movement at specific velocities, the stepper motor provides the finest

movement range in all the motor types '2tepper $otors provide angular motion at specific steps

depending on the number of poles in the stator and the rotor of the motor. "he same numbers of pole pairs

are re#uired on the rotor as on a single stator. "he angle of one step is e#ual to

Step Angle=NsNrNs Nr

360

WhereB

=s G =umber of teeth on stator core. =r G =umber of teeth on rotor core.

"hese motors are #uite heavy as more accurate motors would re#uire more poles and a high tor#ue

producing motor would need more powerful magnetic field. "hus better performance would result in

additional weight which we are trying to reduce in the mobile part of the manipulator body.

"hese motors re#uire a system which includes a ontroller, Amplifier and the motor itself ). "he controller

needs to be able to produce 3W$ ?i.e. pulses@ and an 1-bridge circuit would be re#uired to produce

movement in both clockwise and counter-clockwise motion. "his would work as an electrical switch to

change direction which can be written in the programming instead of a mechanical switch which would

re#uire mechanical interference to change the direction of motion. "he Amplifier increases the power of thepulses so that the motor can be powered up by it. As not all amplifier are compatible with stepper motors so

they need to be compatible with each other to control the motor properly. Finally, the stepper motor is

needed to convert the power in the amplified pulses to mechanical work.

tepper motors are relatively cheap, produce a high tor#ue at low speeds and work in all sorts of

environments thus they are #uite reliable as they provide what is needed from them but at high speeds, the

tor#ue produced is significantly reduced and resonance might occur at low speeds.

"he specifications re#uired to choose the motor are as follows 1+11B

*. Dynamic "or#ueB "or#ue developed when in motion.

. 3hase InductanceB Eimits phase-current rise time and thus Dynamic "or#ue.

6. 1olding "or#ueB "or#ue to prevent involuntary motion by static load.

7. "or#ue tiffnessB $otor9s ability to resist angular displacement within a step.

5. >otor InertiaB >esists motor accelerations.


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"here are many types of configurations in which a motor can be placed in a manipulator and these have their

advantages and disadvantages over their counterparts. "he following is a list of configurations which have

been used in serial planar manipulators. "ransmission systems are used to transfer the power from the

$otors to the &oints so that the rotary movement in the motor can be converted to the specific movement

re#uired at the joint.

$otor at Eink transmission system.

"endon - based transmission system.

/elt - based transmission system.

0ear - based transmission system.

3neumatics21ydraulics - based transmission system.

A detailed e;planation of each of these transmission systems is given below in the order shown above.

"he tor#ue re#uired for the manipulator would be based on the weights of the link and the joints which include

the transmission ?depending on the design@ and motors.

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,I12+5 The link diagram o. a 3 l ink mani/ulator2

m*, m and m6 are the masses of the links and the transmission system parts which go along the links. "he link

weight is taken to be acting in the centre of the link as we are assuming the mass is evenly distributed.

All of the designs shown above would have different masses depending on the system they are using and they

would have limitation of the tor#ue which they would be able to transfer.

*onventional $ani/ulator

,I12+ A mani/ulator with motors at the base2 13

In this design, the motors will contribute to the joint weight and as

there is no transmission system as the motor is directly connected to the

link thus there is no additional inertia which the motor would have to overcome when moving its respective

link.

"he motor is the heaviest part of the manipulator and thus the motor at the previous joint would have to be

more powerful as it would be supporting the weight of the motor and thus higher tor#ue would be re#uired

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thus the parts in that motor need to be more powerful thus a larger motor is re#uired and thus consecutive

motors would be larger than the previous.

"he motors in this design would get smaller with each consecutive link thus each joint would provide a

smaller tor#ue thus if a larger tor#ue is re#uired at that joint, the previous motor would be re#uired to be

larger even though the higher tor#ue is not needed over there and is only need due to the additional weight of

the larger motor.

,I12+# A sim/ler illustration o.

a conventional mani/ulator2 13

H

Tendon 8 based transmission s%stem

,I12+' A sim/le tendon based design2

"his model uses tendons to transfer the force from the motors to the links as shown in FI0*.+%. It converts

a;ial displacement to rotational displacement as the tendons are pulled from each direction to rotate the link

in the direction of the pulling force. "he force is not applied at the joints but at a distance from the joints thus

a tor#ue is applied at

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Force ; os?@ ; is?J@?y-a;is@

Force ; in?@

Force applied

by tendon


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,I12+) The .orce s/lit into its com/onents2 13

"he force which would create the rotational motion

would be the y a;is component of the force on the tendon. "he ;-a;is component of this force would act at the

joint which would increase the friction between the link bars and the joint rod as the force and this would cause

hindrance to the motion of the manipulator. "his is called the joint stiffness.

,I121+ The e..ect o. .riction on this s/eci.ic

design2

$ore than one actuator ?or larger actuators@ can be used for joints without changing the dynamics of the

manipulator.

"hus the total force produced by the motor would not be utili8ed to produce the rotational motion. "he efficiency

of the transmission system is fairly low but this is balanced by the reduced weight at the links as the motors are

replaced from the links to the base of the manipulator.

,I1211 Tendon8Pulle% (esign One2013

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?@

J?;-a;is@ Force ; os?@

K. Force ; os?@ ; os?J@

Friction G LMFcos?@&oint >od

Force ; os?@

Eink /ar


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Number of Pulleys=Number ofJoints

In this design, the tension of the taut wire produces the force which causes friction between the tendon and the

pulley and hence the tendon is able to transmit the tor#ue from the motor to the joint. As the tendon for the joint

6 is passing through the pulleys * and , there is a tor#ue produced in the *stand ndjoint thus there would be

unnecessary motion at these joints thus, when programming the controller for this joint, there must be a feedback

system which would have to compensate for the irrelevant motion thus the programming would get more

complicated and hence the flaw in the design.

,I121& Tendon8 Pulle% design two2013

"his design tries to remove the irrelevant motion by using two pulleys instead of one for each tendon at the

joints where motion is not needed. Instead of the tendons stretching at opposite sides of the pulley, they are now

placed on the same side of two pulleys thus compensating for the motion of each other.

,I1213 The conce/t o. the design2013

Number of pulleys=(number of joints)2

As the pulleys increase, the overall weight of the manipulator increases. "his additional weight would not hinder

the manipulators movement at a small scale. As we increase the si8e of the manipulator, a larger tor#ue would be

re#uired to move the manipulator hence increasing the force thus increasing the shear on the pulley. As shear

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"or#ue ?clockwise@

"or#ue ?anti-clockwise@


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increases, the material with lower shear modulus would deform under the force "his would be conflicting with

the design itself as the design was to decrease the

As we increase the si8e of the manipulator, a larger tor#ue would be re#uired to move the manipulator hence

increasing the force thus increasing the shear being applied on the pulley. As shear increases, the materials with

lower shear modulus would deform under the force thus materials with higher shear modulus are re#uired.

According to FI0*.*7, the density of the materials increases linearly with increasing shear modulus thus for

higher tor#ue application, this design would not be as effective as the weight would rise sharply.

"he width of each link would increase with the increase of joints of the manipulator thus this would lead to abulky design which is not ideal for most applications.

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Force"his is one instance of the

force on the pulley. "here

would be forces tangent toevery point of

circumference of the

pulley.

"his parallel force causesdeformation along the surface of the

shape.

3ulley


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-ensit) 2kg m435100 1000

earmouus

a

031

1

10

100

,I1214 (ensit% o. $aterial vs Shear $odulus2

Instead of using pulleys, a single ball bearing could be used instead. "his would decrease the bulkiness of the

design and at higher tor#ue, the weight of this design would be less than the double pulley design. As seen in

FI0*.**, there would be a bearing at each non-tor#ue transmitting pulley.

,I1215 A diagram o. a bearing2 14

"he bearing in this mechanism is used to isolate the rotational motion of one part from another and it decreases

the friction between them significantly.

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uter race

Inner race

/alls

age


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"his is caused by the mechanism which includes simple spheres, two races and a cage which holds the spheres in

their place as shown in FI0*.*5. "he spheres i.e. balls come in contact with both the races. "he race which is

connected to the mobile part would be moving at the same velocity and thus the point of the ball in contact with

the race would also move at that velocity thus causing rotation. "he other race would be attached to a stationary

object. "hus the balls would be revolving around the a;is of the bearing. As the balls a rolling, they have a

significantly lower co-efficient of friction and thus it would seems like that the two objects are sliding besides

each other.

9ame o. $echanism -eight :Small Scale;

Tendon Eowest

Tendon8Pulle% Eower ?In-effective design@

Tendon8(ouble Pulle% 1igher

Tendon87earing Pulle% 1ighestTA712+& $echanism vs -eight

9ame o. $echanism -eight :!arge Scale;

Tendon Eowest

Tendon8Pulle% Eower ?In-effective design@

Tendon87earing Pulle% 1igher

Tendon8(ouble Pulle% 1ighestTA712+& $echanism vs -eight

When compared to the conventional design, the highest payload ?when the arm is hori8ontal i.e. the worst case in

terms of gravitational support@ of the designs shown above are significantly higher when the same driving

system is used thus only changing the transmission. "his is proven in CDesign and 4;periment of oupled

"endon $anipulators thus these design are more efficient as they provide more useful output tor#ue at the same

input tor#ue as the weight of the manipulator is decreased.

"he material used for the tendons is also an issue as for higher tor#ue applications, the material needs to have

higher tensile strength and thus higher (" so that the tendon would not break during the application or deformplastically.

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Tensile strengt" 2,Pa501 1 10 100 1000

Pri

ce2,7'3kg5

10

100

1000

Polyester4E-lass ber6 non-crm' abrc com'oste6 uas-sotro'c lamnate

Polyester45wt% E-lass ber6 wo!en abrc com'oste6 ba+al lamnate

Polyester4E-lass ber6 'ultru&e& com'oste ro&6 un&rectonal

,I121 The gra/h o. Tensile Strength vs Price2

3olymers are normally used for tendon materials as they are fle;ible thus they are able to bend easily ."hus the

best material would be polyester24-glass fiber as this would be under $!>*+2kg whereas Nevlar which is

usually chosen for tendons which costs around $!>


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7elt < 7ased transmission s%stem2

"his transmission system has the same concept as the tendon- based transmission. Instead of using tendons and

pulleys, belts are used with pulleys and gears. /elt drives can absorb shock and vibrations on the system as wellas a degree of misalignment in the driver and the driven object.

/elts can be of two types, flat or :ee beltsB

Flat beltsB"hese belts provide high power at high speeds when the scale of the transmission is large so the

system is bulky and thus are suitable for small systems which re#uire large tor#ue. "hese belts re#uire large

amount of tension to transmit power properly which increases the load proportionally. "hese belts tend to move

about the a;is of rotation so a centering mechanism is used to keep them at place. As the force which makes it

possible to transmit the force from the belt to the machine element is friction aloneO at high speeds and when thesystem has large inertiaO slippage would occur, this would not be suitable. "he material used in these belts need

to have a high friction co-efficient so that the grip of the belt is improved so it would be able to transmit theforce efficiently.15

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,I121# A t%/ical .lat belt2

:ee beltsB "hese belts have a different shape as wedges are created along the shape of the belt which would give

them better hold on the machine element which it is connected to.

,I121' A =ee :Timing; belt2 1#

"hese belts provide the best dynamic characteristics i.e. the traction force, speed of movement and load on

bearing. "he shape of the e;trusion on the belt is trape8oidal as this would provide a good angle of approach and

e;iting angle and the edges of the shape are rounded so that the force does not cause wear on the belt due to

constant shear. And thus due to the shape of the belt, the lifetime of the belt is also prolonged.

"he material of these belts could have a lower frictional co-efficient as the majority of the force is transmitted as

contact force as shown in FI0*.*'.

,I121) The .orces between a =ee belt and a ear2

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"he downward

force from the belt

acting on the gear.

Force from belt directed on the teeth of the gear

Force from belt transferred due to friction

between the two machine elements.


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"he overall system for the belt2gear and belt2pulley are similar to the tendon2pulley mechanism.

ear 8 based transmission s%stem2

0ears are the most common machine elements used to transfer power. "hus there is a couple of different type of

gears used in manipulator transmission.1#

*. 3arallel A;isB "hese include spur gears and helical spur gears and their teeth are perpendicular to thesurface. "hey are used to connect to a;is of rotation.

,I12&+ Two gears connecting their a0es2

. 3erpendicular A;isB "his is further classified into Intersecting A;is ?traight and piral /evel 0ear and

Face 0ears@ and =on-Intersecting A;is ?Worm 0ear and 1ypoid 0ear@ which e;plains by itself.

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,I12&1 7evel gears connecting their a0es2

6. rossed A;isB "he a;es of these gears are neither parallel

nor perpendicular to each other.

,I12&& Two helical connecting their a0es2

7. 3lanetary 0ear ystemB

,I12&3 A Planetar% ear with all the elements labelled201%

f these gears, the perpendicular gears are used in the transmission system due to the transfer of power in

perpendicular direction to the joint rotation a;is.

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,I12&3 The mechanism used in Samer >ah%a et al2 in their /a/er201(

In amer !ahya et al.9s design, all the joints were controlled by a single motor. "his would thus create the same

angular movement at each joint and individual joint movement is not possible in this design. "he overall tor#ue

usage of this design was less than of the conventional manipulator design.

/evel 0ears vs Worm 0ears for transmitting power at a right angleB&+

According to iemens research team, the bevel gears are more efficient at transmitting tor#ue and have a longer

life-time than their counterpart, the worm gear.

,I12&4 The advantages and disadvantages o. di..erent t%/e o. gears2

Column1 8elical ear 9e!el ear

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earbo: ;


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$ani/ulator (esign and Pro ngineer $odelling

$any designs were thought of in the start and during the period of the project. f those designs only a couple of

designs were selected.

"he following designs were selected on the basis of their mechanism and the ability of 3ro-4ngineer to model

and simulate them.

The 7evel8ear transmission s%stem2

"his mechanism utili8es the ability of the bevel gear mechanism to transfer tor#ue at a right angle and converts it

into a mechanism which can transfer tor#ue from a source to any angle via a link. "his mechanism uses bevels

gears and bearings for conveying tor#ue to the links and not allowing the transmission system to create motion

in unnecessary links respectively. "his design was modelled after understanding the design of amer !ahya et

al.9s design in their paper C&oint "or#ue >eduction of a "hree Dimensional >edundant 3lanar $anipulator. A

few modifications were made on it as the design used by them utili8ed a single motor which would control all

the links motion thus individual link control was not feasible.

7evel ear $echanism

"he mechanism is basically set on a single a;is of rotation and the bearing creates relative motion of the link

shaft and the bevel gear so the rotation of the gear does not affect the rotation of the shaft. "his gear is connectedto another gear perpendicular to it thus this makes the bevel gear.

Page2(

:ertical A;is *

:ertical

A;is

/evel 0ear 7

/evel 0ear

6


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,I12&' The design o. the mechanism2

17evel ear 1B&B3B4 "he bevel gear ?7 bevel gears as shown@

&"he bearings ? as shown@

3"he hollow shaft connecting the gears.

4"he gear holder.

5"he a;is of rotation of joint rod, bearings, the gear connecting hollow shaft.

"he bevel gears 1 and &are connected to each other by gear meshing as are the bevel gears 3 and 4. "he design

does not allow the transfer of tor#ue from the gears to the joint shaft which would be inserted in between the

bearings. "his is done by the action of the bearings which stops the transfer of tor#ue from the gears to the joint

shaft. Part3is there so that the bevel gears & and 3are able to transfer tor#ue between each other. If there was

not Part3, the tor#ue would not be transferred and the two bevel gear systems would separate systems which

would be connected by the joint shaft which would have been inconvenient. Part 4 is placed so that the bevel

gears 1 and 4do not move in the direction of the a;is 5O the a;is of the joint shaftO thus they would always be

fi;ed in their position so that there are no obstacles when moving the manipulator.

Instead of the two bearings, a single bearing can be used instead as it would provide the same function.

"here are three joints in the manipulator designed thus there are < bevel-gear mechanisms in the manipulator.

Page3/

/evel 0ear

/evel 0ear


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,I12&) The design bevel8gear transmissions%stem2

"he bevel-gear mechanism shown in FI0*.' is only for joints where the tor#ue from the motor is not needed

thus there are of these mechanisms on &oint * and * on &oint .

Page31

/evel 0ear 1ori8ontal A;is

:ertical A;is

/evel 0ear *


32/153

,I123+ 7evel ear $echanism :ear to 6oint Sha.t TorCue Trans.er;

"he mechanism shown above is for transferring the tor#ue from gear 1to gear &which is connected to the joint

shaft thus effectively transferring the tor#ue from the motor to the joint shaft which was meant to be accelerated.

"here is * of these mechanism on each of the joints as which these mechanisms, the manipulator would be

unable to move as no tor#ue transfer would be taking place.

"hese mechanisms are connected to their consecutive mechanisms via shafts which would connect the verticala;is of one mechanism to the vertical a;is of the ne;t mechanism thus each of the bevel gears in one line would

be connected to each other and would move with the same rotational velocity as the gear ratio of the whole

mechanism is *B*.

Due to the gears being of *B* ratio ?i.e. the same si8e@, the a;ial distance of the manipulator would be large. "o

reduce this, the gears on the hori8ontal a;is need to be larger than the gears on the vertical a;is thus the a;ial

space needed to place all the mechanism on the joints would be less thus all the vertical a;is gear would be

smaller compared to the hori8ontal a;is gears at each joint thus effectively reducing the a;ial space occupied by

the manipulator.

"his mechanism could hence be used in manipulator with more joints as the a;ial si8e of the manipulator can be

controlled instead of increasing with each additional joint. "he manipulator base is connected to the stationarybase via a bearing thus allowing the whole manipulator to be able to rotate thus giving it freedom in 6

dimensions.

"he materials used in this design were used in order to reduce the mass of the overall design thus reducing theweight and hence the overall tor#ue needed to move the manipulator while trying to not change the part

properties so much that the parts would fail while in motion.

"he parts for this design were as followsB

Parts ,aterials -ensit) kgm@*,nAs $e-.ar&ene& wrou.t alumnum 2900

Bransmsson Ro& ,ow Carbon teel 7850Gear $e-.ar&ene& wrou.t alumnum 2900Gear ;ol&er $e-.ar&ene& wrou.t alumnum 2900


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7oung>s modulus 26Pa51e-5 1e-: 03001 0301 031 1 10 100

-ensit)2kgm

435

100

1000

10000

As the manipulator is not very huge, the tor#ue re#uired to move it would be not huge thus aluminum could be

used without the fear of deformation.

"he materials for a manipulator are directly related to the si8e of the manipulator as a larger manipulator would

re#uire an overall larger tor#ue and thus would need gears which are re#uired to transfer larger tor#ue so the

material of the gear would need to have a larger young9s modulus and thus the density of the material would

increase ?as shown in the graph below@.

,I1231 !inearit% o. (ensit% and >oungDs $odulus2

As 3: can be used to create a small manipulator while a large scale manipulator would fail if it uses the samematerial as the small manipulator.

Pro*erties ?nits&ow Carbon

$teelAge@"ardened wroug"t

aluminum

Prce >R4A 18 75

el& tren.t >Pa 510-10 300-395Bensle tren.t >Pa 550-20 500-580

Com'resscetren.t

>Pa500-10 300-395

ouns >o&ulus G'a 8-80 200-215

Castablty


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>ac.nablty Goo& ery Goo&

Fel&ablty ery Goo& Goo&

TA712+5 Pro/erties o. Aluminum and Steel2

!imitations

As said, the designing of the model was supposed to be done on olid works but due to the license of olid

works not being available to the university, the modelling was done in 3ro-4ngineer.

3ro-4ngineer in the university did not include the module in 3ro-$echanism which defines bevel gear

connection as it had only standard and rack and pinion connection.

,I123& T%/es o. ear *onnections available in Pro8$echanism2

Tendon8

Pulle%

Transmission

S%stem

Page34


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,I1233 The basic design o. the Tendon $ani/ulator2

"he basic concept of this manipulator is derived from the fact that tendons can transfer force and pulleys can be

used to change the direction of the force.

In this, the tendon wraps around the joint pulley ?i.e. the tendon goes around the pulley for 6


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Pro*erties ?nits e!lar 14(@aramid Bber

Densty A4m@ 1:70Prce >R4A :00

Bensle trent. >'a 3:00ouns >o&ulus G'a 180

Possons Rato 03

#le+ural >o&ulus GPa 180Elonaton % stran 12

>ec.ancal ,oss Co-ecent 0008-001TA712+ Pro/erties o. Eevlar 14)8aramid .iber2

"his design would produce a manipulator which would achieve a light-weight design thus it would have a low

moment of inertia. "his would then re#uire low energy consumption thus improving efficiency. As the weight of

the manipulator is less, the pay-load to weight ratio would be higher.

Parts ,aterials -ensit) kgm@*,nAs $e-.ar&ene& wrou.t alumnum 2900

Bransmsson Ro& ,ow Carbon teel 7850


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*alculations

"he calculations for the manipulator used the Denavit-1artenberg parameters.&4&5&

"he manipulators designed had 6 links thus 6 DFs and thus 6 joints in the D-1 parameter table. All the joints in

the manipulator are revolute

+oints D a d +oint 'ange

1 )1 0 l1 0 (-*90H to 85H

2 )2 0 l2 0 (-*90H to 90H

3 )3 0 l3 0 (-*90H to 90H

TA712+' (8@ Parameters .or the $ani/ulator2

Parameter

-eBnition

EiF Rotaton anle rom I-1to Iabout J-1EDiF $nle o rotaton rom J-1to Jabout IFaiG Distance from intersection of Pi-*Q Rito the origin of i coordinate along Ri.

EdiF Dstance rom orn o (-1* coor&nate to ntersecton o J-1K IalonJ-1

TA91G/(H -eBnitions of t"e -@8 ParametersG

.

Page3#

8-a;is


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,I1235 The design o. the mani/ulator2

Page3%

l6

l

l*rigin ;-a;is

y-a;is


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,I123 Einematics Parameters o. a $ani/ulator2

With the D-1 parameters, we need to find the tor#ue e#uation which is given as follows.

=D ( ) (t)+h ( , )+c ()

In this e#uationB

G "or#ue of the $anipulator

D ( )=The Accelertionrelte! "nerti

h ( , )=The #entrifugl#oriolis

Page3(


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c( )=The$r%ity term

,inding the Acceleration related Inertia $atri0 Term

D ( )=

(D 11 D 12 D 13

D21 D 22 D 23D31 D 32 D 33)

O D?i j@

4ach of these terms is the value for the tor#ue of the jthlink transferred to joint i.

4ach term of D?i j@ is given as follows ?where j is replaced by k@B

If iGk, it will be Diiwhich will be related to the acceleration of joint i where the driving tor#ue " iacts while

when i is not e#ual to k, Dikis related to the reaction tor#ue induced by the acceleration of joint k and acting at

the joint i or vice versa.

Where &iis the 3seudo-Inertia $atri; of link i, dependent on the mass distribution of link i and are e;pressed

with respect to the i thframe.

Where (jk is the effect of motionof joint j on all the points on the

link i.

Page4/


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Where (LAB s t.e trans'ose o t.e MLA w.c. s !en as ollows

Trnspose

(1 2 3

b1 b2 b3

c1 c2 c 3)=

(1 b1 c1

2 b2 c2

3 b3 c 3)"he i-*"iin the previous e#uation is given by the following e#uationB

"his is the same for all rotary joints.

"he Sj isgiven by the following matri; for rotary jointsB

&i=(0 1 0 01 0 0 0

0 0 0 0

0 0 0 0)"hus the D-1 parameters are inputted in the i-*"i and thus by following the procedure shown above, the

Acceleration related Inertia matri; can be found.

ince the Inertia matri; is symmetric and "r?A@ G "r?A"@, DikG Dki.

,inding the ravit% "elated $atri0 Term

Page41


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c ( )=c 1

c 2

c 3

4ach of these terms is given by the following e#uationB

Where

iriG (l

2

0

0

1

) which is the displacement of point I from the link co-ordinate frame i.

i G gravity loading term due to the links

,inding the *entri.ugal and *oriolis $atri0 Term

h ( , )=(h1h2h3

)Where hiis found out by the following termB

Where hikmis given by the following e#uationB

Where (jkmis given by the following e#uationB

Page42


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Where "r? @ is the trace of the matri; which is the sum of the main diagonal of the matri;.

Tr (A )=( 1 2 34 5 67 8 9

)=1+5+9"he co-efficient hikmis related to the velocity of the joint variables and is defined by the e#uation given above.

"he two indices k and m are related to the velocities of joint k and m, whose dynamic interplay induces a

reaction tor#ue at joint i. "hus, the first inde; I is always related to the joints where the velocity-induced reactiontor#ues are felt.

For kGm, hikkis related to the centrifugal force generated by the angular velocity of joint k and felt at the joint i,

while when k is not e#ual to m, hikmis related to the oriolis force generated by the velocities of joint k and m

and is felt at joint i.

"hus, the entrifugal and oriolis $atri; term can be found and now the tor#ue values for each joint can be

found out.

ome of these co-efficient can be 8ero. "his can be due to the following reasonsB

*. "he kinematic design of the manipulator can eliminate some dynamic coupling between joint motions.

. ome of the velocity-related dynamic co-efficient only have a dummy e;istence i.e. in the entrifugal

and oriolis related $atri; "erm.

6. Due to particular variation in the link configuration during motion, some dynamic co-efficient may

become 8ero at particular instance of time.

Page43


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'esults

"he manipulators were simulated on 3ro-4ngineer to find out the tor#ue re#uired for the manipulators at each

angle.

0ravity and all the friction between the parts were taken into account during the simulations. "he tor#ue re#uired

at the joints was found out every joint for the manipulator

In the simulations on 3ro-4ngineer, the following simulations were done to find out the tor#ue re#uired for the

manipulator without any end loadB

*. "he tor#ue re#uired when the $anipulator is at its hori8ontal position ?i.e. the position which re#uires

the ma;imum tor#ue@.. "he tor#ue re#uired when the $anipulator is vertical position. ? the minimum tor#ue re#uired at the

joints@

6. "he tor#ue re#uired when the joints of the manipulator are moving from their ma;imum angular position

to minimum position and vice versa. All of them move at the same angular velocity.

7. "he tor#ue re#uired when the *stlink is hori8ontal while the ndand the 6rdlink are rotating at the same

velocity.

5. "he tor#ue re#uired when the *stand the ndlink are hori8ontal while the 6 rdlink is rotating at a given

velocity.


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). "he tor#ue re#uired when the *stand the ndlink are vertical while the 6rdlink is rotating at a given

velocity.

"hese simulations are done for each of the manipulator i.e. the bevel gear manipulator, the tendon manipulator

and the conventional manipulator.

>esults for 7-) are given in the appendi;.

0 01 02 03 0: 05 0 07 08 09 1

0

05

1

15

2

25

3

TorCue ,or 6oints (uring @oriHontal Position

ont 1

,near (o

ont 2

,near (o

ont 3

,near (o

Time?sec

TorCue :9m;

7evel ear $ani/ulator

Page45


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0 01 02 03 0: 05 0 07 08 09 10

0

0

0

0

00

0

0

+oint Torue w"en t"e ,ani*ulator is *er*endicular to t

Time

Tor=ue

,I123# TorCue at 6oints reCuired to sta% at the horiHontal /osition2

,I123' TorCue reCuired .or being still during the =ertical /osition.

Page46


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-1 1 3 5 7 9 11 13 15

-3

-2

-1

0

1

2

3

Nont 1 Nont 2 Nont 3

Time? Seconds

TorCue ? 9m

Page4#


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,I123) TorCue vs Time .or rotating joints

All the joints were rotating at a steady *+ degree2second speed. "hus the ndjoint had a speed of +

degree2second and the 6rdjoint had an angular speed of 6+ degree2second.

0 001 002 003 00: 005 00 007 008 009

0

02

0:

0

08

1

12

1:

1

18

Time? Second

Tor=ue3Im

Tendon $ani/ulator

,I124+ TorCue reCuired .or the mani/ulator .or it to sta% in its horiHontal /osition2

Page4%


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0 001 002 003 00: 005 00 007 008 009 01

0

0

0

0

0

0

0

0

0

ont 1

ont 2

ont 3

Time? Seconds

TorCue? 9m

,I1241 TorCue reCuired .or the $ani/ulatorDs vertical /osition2

Page4(


50/153

,I124& TorCue vs Time .or rotating joints2

0 2 : 8 10 12 1: 1 18

-15

-1

-05

0

05

1

15

Non

Non

Non

Time? Second

TorCue? 9m

Page5/


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*onventional $ani/ulator

0 01 02 03 0: 05 0 07 08 090

1

2

3

:

5

7

8


Time? seconds

TorCue? 9m

Page51


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0 1 2 3 : 5 7 8 9 100

0

0

001

001

001

001

001

Time? seconds

TorCue? 9m

,I124& TorCue .or @oriHontal Position2

,I124& TorCue reCuired .or the =ertical Position2

Page52


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0 2 : 8 10 12 1: 1

-8

-

-:

-2

0

2

:

8


TorCue ? 9m

Page53


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,I1243 The TorCue vs Time gra/h .or rotating joints2

*om/aring the $ani/ulator (esigns

,or @oriHontal Position

0 001 002 003 00: 005 00 007 008 0090

05

1

15

2

25

3

Con!entonal $rm Ben&on $rm


55/153

Page55


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0 2 : 8 10 12 1: 1 18

-8

-

-:

-2

0

2

:

8

an'ulator Con!entonal >an'ulator Ben&on >an'ulator

Time ? Seconds

TorCue ? 9m

,or "otating 6oints

,I1245 The TorCue .or 6oint 1 .or each $ani/ulator2

Page56


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Page5#


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,I124 The TorCue .or 6oint & .or each $ani/ulator2

Page5%


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0 2 : 8 10 12 1: 1 1

-25

-2

-15

-1

-05

0

05

1

15

2

25


TorCue? 9m

Page5(


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Page6/


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,I124# The TorCue .or 6oint 3 .or each $ani/ulator

Page61


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0 2 : 8 10 12 1: 1

-03

-02

-01

0

01

02

03


TorCue ? 9m

Page62


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-iscussion and Anal)sis

As seen from FI0*.77 which displays the tor#ue re#uired by the manipulator to be in the hori8ontal position

which is shown below. "his position re#uires the ma;imum tor#ue, as the moment of each link is highest as

gravity is acting perpendicular to the hori8ontal links, thus the hori8ontal distance from the centre of mass of the

link to the link joint is ma;imum creating the largest tor#ue ?momentum@.

,I124'

The

horiHontal /osition o. the mani/ulators2

"he manipulator above is the bevel gear manipulator shown in the hori8ontal position.

"he vertical position is perpendicular to this position as such gravity would be acting parallel to the vertical links

thus the hori8ontal distance from the moment of the link to the joint would be minimum.

"he rotation of the manipulator was done between the two e;treme positions.

"he results we got from the simulations showed us that manipulators with the transmission systems would

re#uire less tor#ue at the links to create the same motion as the conventional manipulator. "he tor#ue re#uired by

the manipulator is directly proportional to the mass of the manipulator. As the mass of the manipulator increases,

the tor#ue increases thus in a larger scale of the manipulator where a larger motor is needed the difference in thetor#ue re#uired would diverge further and the transmission system would yield a higher efficiency as the base

manipulator movement would re#uire less input power as less tor#ue is needed to move the manipulator.

"hese transmission systems can be implemented in planar manipulators with more DFs as we have discussed

how to decrease the a;ial si8e ?width@ of the manipulator so as the links increase the a;ial si8e does not increase

proportionally.

Page63

&o

&o

&o


64/153

"hese manipulator designs would be able to use the same motor si8e at each link whereas the conventional

manipulator would re#uire a smaller motor with each increasing link as the previous motor needs to support the

ne;t motor.

1aving the same motor si8e at each link in the conventional design would not be logical as the base tor#ue

re#uired to move the manipulator would increase drastically.

Conclusion

"he main objective of the project was to reduce the joint tor#ue of a manipulator. For this, e;tensive research

needed to be done to find out the types of methods to reduce the tor#ue re#uired at the link for a manipulator.

"he method to reduce the joint tor#ue of a manipulator which was chosen was the use of a transmission system

and usage of lighter materials so that the overall design had a lower mass. "he manipulators were designed with

their respective transmission systems.

"hese were then modelled in 3ro-4ngineer. Dynamic simulations were then run on these manipulators using the

mechanism module of 3ro-4 and by giving re#uired constraints, the tor#ue was found out for each of the joints

of the manipulators.

Further calculations were done to show the dynamic model of the planar manipulator. "his helped in

understanding the manipulator design and the effect of Inertia, 0ravity, the entrifugal and oriolis 4ffect on the

manipulator when it is accelerating, moving at a constant speed or stays at rest.

"he manipulators designed were then compared with the conventional manipulator design. "he difference in the

tor#ue was seen as the conventional manipulator had an overall larger joint tor#ue re#uirement than the

manipulators using the transmission systems.

It was seen that the bevel gear transmission system would have a higher tor#ue re#uirement than the tendon-pulley transmission system but both of these manipulators had an overall less tor#ue re#uirement than the

conventional design.

All aims and objectives which were stated for this project were fulfilled.

.urt"er Jork

Due to the unavailability of olidworks, the simulations of the design of the tendon were done on 3ro-4ngineer

so the tendons were not added in the simulations and only the joints were simulated with the overall parts. "he

simulation can be done with the tendons attached in the simulations to get more accurate results.

Page64


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"he designs can be created in hardware and then simulated to get real worlds results as the budget was

insufficient as the tendon and bevel gear manipulator combined would have cost way above the given budget of

$!> 6++.

'eferences

T*U. Dennis ui. ?+*@. "he un9s the limit. Young Scientist Journal. 5 ?*@, oboticsVNinematicsVandVDynamics2erialV$anipulatorV3ositionVNinematics.

TAccessed *5 April *7U.

T


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T*U. Indian Institute of "echnology Nanpur. +*7.IN&R("1C&I(N &( S'RIA2 ARM. T=EI=4U Available atB

httpB22home.iitk.ac.in2adutta2$anualVserialVarm.pdf. TAccessed *5 April *7U.

T*6U. hugen $a, higeo 1irose, and 1iroshi !oshinada , .$, .1, 1.!, *))6. Design and 4;periments for a

oupled "endon-Driven $anipulator . Control S#stems I''', *62*, pg6+-6eduction of a

"hree Dimensional >edundant 3lanar $anipulator . Sensors 5Basel6, *2. 0on8ale8, ..0 Eee, *)'%.Robotics/ Control, Sensing, -ision and Intelligence. *st ed.

InternationalB $c0raw-1ill International 4dition.

T5U. $an Phihong. >obotics For omputer 4ngineering tudents. 6rded. InternationalB3earson 3rentice 1all.

Tobotics, Dynamics and ontrol. nded. International. &ohn Wileys and

ons.

T%U. >achid $anseur. >bot $odelling and Ninematics. *st ed. International. Da :inci 4ngineering 3ress.

Page66
http://en.wikibooks.org/wiki/Robotics_Kinematics_and_Dynamics/Serial_Manipulator_Position_Kinematicshttp://en.wikibooks.org/wiki/Robotics_Kinematics_and_Dynamics/Serial_Manipulator_Position_Kinematicshttp://www.coppeliarobotics.com/helpFiles/en/jointDescription.htmhttp://www.comsol.com/multibody-dynamics-modulhttp://www.comsol.com/multibody-dynamics-modulhttp://www.haydonkerk.com/Resources/StepperMotorTheory/tabid/192/Default.aspxhttp://www.haydonkerk.com/Resources/StepperMotorTheory/tabid/192/Default.aspxhttp://yertiz.com/cnc/steppermotor.pdfhttp://www.embedded.com/print/4378311http://www.coppeliarobotics.com/helpFiles/en/jointDescription.htmhttp://www.comsol.com/multibody-dynamics-modulhttp://www.comsol.com/multibody-dynamics-modulhttp://www.haydonkerk.com/Resources/StepperMotorTheory/tabid/192/Default.aspxhttp://yertiz.com/cnc/steppermotor.pdfhttp://www.embedded.com/print/4378311http://en.wikibooks.org/wiki/Robotics_Kinematics_and_Dynamics/Serial_Manipulator_Position_Kinematics


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A**endi:

0raphs for >esult 7-). All the graphs have "or#ue on their y-a;is and time on the ;-a;is.

"he se#uence which will followedB

7. &oint * is kept in the hori8ontal position while &oint and &oint 6 rotate through their joint rangeB

Page6#
http://www.pfeiferindustries.com/timing-belt-pulley-pitch-diameter-outside-diameter-charts-i-12-l-en.htmlhttp://www.carparts.com/transmission.htmhttp://home.iitk.ac.in/~adutta/Manual_serial_arm.pdfhttp://www.impactbearing.com/deep-groove-ball-bearings.htmlhttp://www.pfeiferindustries.com/timing-belt-pulley-pitch-diameter-outside-diameter-charts-i-12-l-en.htmlhttp://www.pfeiferindustries.com/timing-belt-pulley-pitch-diameter-outside-diameter-charts-i-12-l-en.htmlhttp://www.gearsandstuff.com/types_of_gears.htmhttp://www.carparts.com/transmission.htmhttp://www.industry.usa.siemens.com/drives/us/en/energy-efficient-ac-gear-motors/Resources/Pages/Worm-vs-Bevel-Gear-Motor-Comparison.aspxhttp://www.industry.usa.siemens.com/drives/us/en/energy-efficient-ac-gear-motors/Resources/Pages/Worm-vs-Bevel-Gear-Motor-Comparison.aspxhttp://www.festo.com/net/SupportPortal/Files/42076/Festo_Biomechatronic_Footprint_en.pdfhttp://www.festo.com/cms/en_corp/9655_10219.htm#id_10219


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-1 1 3 5 7 9 11 13 150

1

2

3

:

5

7

8

N1

N1

N1

&oint *

Page6%


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-25 -05 15 35 55 75 95 115 135 1550

1

2

3

:

5

7

8

&oint

Page6(


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Page#/


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-1 1 3 5 7 9 11 13 15

-03

-02

-01

0

01

02

03

Page#1


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&oint6

5. &oint * and &oint are kept in the hori8ontal position while &oint 6 rotates through its joint rangeB

Page#2


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-1 1 3 5 7 9 11 13 15 170

1

2

3

:

5

7

8

N1 B

N1 ange.

Page%4


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-1 1 3 5 7 9 11 13 15 170

1

2

3

:

5

Nont 1 Ben&on

Nont 1


86/153

&oint *B

Page%6


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0 2 : 8 10 12 1: 10

02

0:

0

08

1

12

1:

1

18

Nont 2 Ben&on Nont 2


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&oint B

Page%%


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0 2 : 8 10 12 1: 1

-03

-025

-02

-015

-01

-005

0

005

01

015

02

Nont 3 B Nont 3


90/153

&oint 6B

'. &oint * is kept the position that is perpendicular to the ground while &oint and &oint 6 rotate through its

&oint range.

Page(/


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0 2 : 8 10 12 1: 1

-25

-2

-15

-1

-05

0

05

1

15

2

25

N1 B N1


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&oint *B

Page(2


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0 2 : 8 10 12 1: 1

-25

-2

-15

-1

-05

0

05

1

15

2

25

Page(3


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&oint B

Page(4


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-02

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&oint 6B

). &oint * and &oint are kept at a position perpendicular to the ground while &oint 6 rotates through its

joint range.

Page(6


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-03

-02

-01

0

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02

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N1


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&oint B

0 2 : 8 10 12 1

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&oint 6B

7evel ear $ani/ulator (esign

Page1//

$echanism to create motion in

Eink *

$echanism to create motion in

Eink

$echanism to create motion in Eink 6


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*alculations

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7evel8ear $ani/ulator (rawings

Page141


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Tendon8Pulle% $ani/ulator (rawings

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Documents

Joint Torque Reduction of a Planar Redundant Manipulator Thesis