anadi anant jain

8/18/2019 Anadi Anant Jain

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NAME: ANADI ANANT JAIN

ROLL NO: 2K12/EC/024

REGN. NO: DTU/12/1224

GROUP-D2


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INDE

X

S.no List of Experiments

1 To study various Spatial Descriptions and Transformationsusing Peter Corke Toolbox in MATLA!

" To study t#e $or%ard kinematics of a planar & degree of freedom ''' robot using Peter Corke toolbox

& To study t#e (nverse )inematics of a planar & degree of freedom ''' robot using Peter Corke toolbox

To perform t#e $or%ard )inematics of a & D*$ planarrobot+ Stanford manipulator+ P,MA robot for t#e given D-parameters using

. 'oboanalyser and calculate t#e end e/ector position for agiven set of 0oint values using MATLA and verify t#eresults %it# roboanalyser output!

To perform t#e (nverse )inematics of a & D*$ planar robotfor t#e given

D- parameter using roboanalyser soft%are and verify t#esolutions from t#e MATLA program!

2 To study velocity kinematics for planar &' robot

3 To study t#e $or%ard Dynamics of & D*$ &' robot usingPeter Corke toolbox!

4 To study Computer 5ision toolbox

6 To study )alman $ilter using MATLA

17 To study *b0ect Tracking using *ptical $lo%!

Page

DateNo.

&

17

14

"&

"6

&&

&3

.1

2&

26

Attendance(Out of 10 classes) Internal Assessment Marks Instructor’s Signature


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Experiment 1

Aim: To study various spatial descriptions and transformations

using Pete r Corke toolbox in Matlab!

88T1 9 se":1+ "+ &7;pi(t represents a translation of :1+ "= and a rotation of &7?%!r!t %orld frame T1 9

7!4227 @7!777 1!77777!7777!4227 "!7777

7 7 1!7777

88trplot":T1+ frame+ 1+ color+ b+ axis+ B7 7 =>T#is option specify t#at t#e label for t#e frame is E1F and it is coloredblue

88#old on88T" 9se":"+ 1+ 7=

T" 9

1 7 "7 1 17 7 1

88trplot":T"+ frame+ "+ color+ r+ axis+ B7 7 =

88#old on>Go% displacement of :"+ 1= #as b een applied %it#respect to frame E1F 88T& 9 T1;T"

T& 9

7!4227

@7!777

"!"&"1

7!777

7!4227

&!4227

7 7 1!777788trplot":T&+ frame+ &+ color+ g+axis+ B7 7 = 88#old on>T#e non@commutativity of composition iss#o%n belo% 88T. 9 T";T1

T. 9


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&


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7!4227 @7!777 &!77777!7777!4227 &!7777

7 7 1!777788trplot":T.+ frame+ .+ color+ c+ axis+ B7 7 =

(a) (i),sing a HIH+ alp#a+ beta+ gam ma+ Juler angle convention+ %rites a Matlabprogra m t#e rotation matrix A R B %#en t#e user enters Juler angles alp#a+ beta+and gamma! J!g! K 9 17+ 9 "7+ 9 &7! Also demonstrate t#e inverse prop erty of rotational matrix %#ic# is B R A

A R B 1 9 : A R B =!

Code:

88 dr 9 piConverting angle in degree to radian

beta 9 ang:"=;drgamma 9 ang:&=;dr TN 9 rotN:alp#a= > 'otate a bout N@axis by angle alp#a Ty 9 roty:beta= > 'otate a bout current y@axis by angle beta Tx 9 rotN:gamma= >'otate about current N@axis by angle gamma'mult 9 TN;Ty;Tx >'esultan t 'otational matrix

Trpy 9 eul"tr:alp#a+beta+gamma= >C#anging euler angle totranslational matrix enter alp#a+ beta and gamma in degreesB17 "7&788 TN

.


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TN 97!64.4 @7!13&2 77!13&27!64.4 7

7 7 1!7777

> Tx Tx 97!4227

@7!777 7

7!777 7!4227 7

7 71!777

7

88 Ty Ty 9

7!6&63 7 7!&."7

71!777

7 7@7!&."7 7

7!6&63

88'mult'mult9

7!31.2@

7!21&1 7!&&24

7!2&&3

7!331

& 7!76.@7!"62" 7!1317 7!6&63

88 Trpy Trpy9

7!31.2@

7!21&1 7!&&24 7

7!2&&37!331& 7!76. 7

@7!"62" 7!1317 7!6&63 7

7 7 7 1!777788'19inv:'mult=

> A R B 1 (nverse 'otational

matrix

'1 9

7!31.27!2&&3

@7!"62"

@7!21&

7!331& 7!1317


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1

7!&&247!76. 7!6&63

88'"9'mult!

> ( A R B )' (nverse 'otational

matrix

'" 9

7!31.27!2&&3

@7!"62"

@7!21&1 7!331& 7!1317

7!&&247!76. 7!6&63

ii)Orite a Matlab program to calculate t#e Juler angles K+ + %#ent#e user enters t#e rotation matrix i!e! A R B !

88 ang 9 tr"eul:Trpy= > C#anging translational matrix to euler angle


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ang 97!13.7!&.61 7!"&2

88ang 'otate about current y@axis byangle beta1 Pa 9 Ty;:Pb=

enter beta in degree"7

enter Pb column %iseB1

7 1

Pa 9

1!"4137

7!633

(b)(i) Write a Matlab program to calculate the homogenous transformation matrix AT B when the user

enters ZYX Juler angles K+ + and the position vector A P B . Test with case K =1! ="! =# and A

P B = $1 " #%&.

88 D'9pi


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7!&1447!4"&"@

7!.264 7@

7!"7.67!.&4 7!41&4 77 77 1!7777

> Tba

Tba 9

7!6".

@7!12&" 7!&."7

1!7777

7!&144

7!4"&"

@7!.264

"!7777

@7!"7.6 7!.&4 7!41&4

&!7777

7 7 7 1!7777

(ii)'or =" ()=! =*! A P B is $# 1%& and

B P = $1 1%&. +se Matlab to calculate

A P .

Pb 9 input:enter Pb column %ise=Pa 9

Tba;BPb1 >Calculating A P using AT B and

B P

Ttrn1 9 transl:&+7+1=

Tba1 9rpy"tr:7+"7;D'+7= Tba" 9 Ttrn1;Tba1enter Pb column %iseB1 7 1

Ttrn19

1 7 7 &7 1 7 77 7 1 17 7 7 1

Tba"9

7!6&63 77!&."

7 &!7777

71!777

7 7 7@7!&."7 77!6&63 1!7777

7 77 1!7777

(iii) Orite a Matlab program to calculate t#e inverse #omogenous transformation

matrix i!e! BT A A T B

1 !


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Also s#o%t#at AT

B

and AT 1 9 I 4 B

88 Tbainv9inv:Tba= eye:.=@ Tba;Tbainveye:.=@ Tbainv;Tba

3


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ans 91!7e@1;

7 7 7!7@

7!1117@

7!7"34 7 7 7@7!7 7!1117 7 7

7 7 7 7

ans 91!7e@1;

7@7!7 7!74&&

@7!111

7@7!71&6 7

@7!7 7

7!74&&7!7 7 77 7 7 7

(c)Write a matlab program to calculate AT C and C T A given the fact that

AT B and

B T C are obtained from part a. ,efine

AT B to be the result from part (a* and

B T C from part b.

> angs"9input:enter alp#a"+ beta"+gamma"= alp#a"9angs":1=;D'beta"9angs":"=;D'gamma"9angs":&=;D'Pcb9input:enter Pcb valuecolumn%ise= Trpy" 9rpy"tr:gamma"+ beta"+alp#a"=

Ttrn"9 transl:Pcb= Tcb 9 Ttrn" ; Trpy" Tca 9 Tba ; Tcb Tac9inv:Tcb=;inv:Tba= Tac"9inv:Tca=

Tba"9Tca;inv:Tcb= Tcb"9inv:Tba=; Tcaenter alp#a"+ beta"+gamma"B17 "7 &7 enter Pcbvalue column%iseB1 " &> Tca Tca 9

7!3&.& @7!766& 7!231 "!2"1


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7!2&3 7!&371 @7!2271"!2 @7!14"6 7!6"&37!&&23 2!&"."

7 77 1!7777

> Tac Tac 9

4


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7!3&.& 7!2&3 @7!14"6@"!..1& @7!766& 7!&3717!6"&3 @2!"2 7!231@7!2271 7!&&23 @"!"71

7 77 1!7777

> Tac" Tac" 9

7!3&.& 7!2&3 @7!14"6@"!..1& @7!766& 7!&3717!6"&3 @2!"2 7!231@7!2271 7!&&23 @"!"71

7 77 1!7777

> Tcb" Tcb" 9

7!6". @7!12&"

7!&."71!77777!&144 7!4"&" @7!.264"!7777 @7!"7.6 7!.&47!41&4 &!7777

7 77 1!7777

> Tba" Tba" 9

7!6". @7!12&"7!&."71!7777

7!&144 7!4"&" @7!.264"!7777 @7!"7.6 7!.&47!41&4 &!7777

7 7 7 1!7777


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Experiment 2

AimQ Orite do%n a Matlab program for t#e planar &@Degree of freedom '''robot %it# t#e

follo%ing parameters L1 9 .m+ L"9&m+ L& 9 "m!To derive t#efor%ard kinematics i!e!

7 T&+ %it#

t#efollo%ing input casesQ9 B7 7 7

T

9 B17 "7 &7 T

9 B67 67 67


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88 L!A:7!=> obtain transformation matrix for t#eta 9 7!radian

ans 97!4332 @7!7777 7!.36. 7!13

7!.36.7!777

7 @7!4332 7!766

71!77777!7777 7!1777

7 7 7 1!777788L!'P

> gets type of 0oint+ ' represent 'evolute 0oint

ans 9'

88 L!a > gets link lengt#ans 9

7!"77788 L!o/set 97! > specify t#eta in advance i!e! 7! rad88L!A:7= > get transformation matrix no%

ans 9> produces same matrix as above as link o/set %asspeciRed initiaily

7!4332 @7!7777 7!.36. 7!13

7!.36.7!777

7 @7!4332 7!766

71!77777!7777 7!1777

7 7 7 1!7777L:1= 9 Link:B7 7 17= > create L:1= Link ob0ectL 9

t#eta9+ d97+

a9 1+ alp#a9 7 :'+stdD-=88 L:"= 9 Link:B7 71 7= > create L:"= Link ob0ectL 9

t#eta91+d9 7+ a9

1+alp#a9 7 :'+stdD-=

t#eta9"+d9 7+ a9 1+alp#a9 7 :'+stdD-=88 t%olink 9 SerialLink:L+ name+t%o link=

> Serial@link robot class represents a serial@link arm@

type robot!t%olink 9t%o link :" axis+ ''+ stdD-+slo%'GJ=

T@@@T@@@@@@@@@@@ T@@@@@@@@@@@ T@@@@@@@@@@@

T@@@@@@@@@@@

U 0 U t#eta U d U a Ualp#aU


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T@@@T@@@@@@@@@@@ T@@@@@@@@@@@ T@@@@@@@@@@@

T@@@@@@@@@@@

U 1U 1U 7U 1U 7UU "U "U 7U 1U 7U

T@@@T@@@@@@@@@@@ T@@@@@@@@@@@ T@@@@@@@@@@@

T@@@@@@@@@@@

grav 97 base 9 1 7 7 7 tool 9 17 7 7

> grav speciRes t#at gravity %ill be applied inN direction

7 7 1 7 7 7 1 7 7 > base speciRes t#e initial position of robot6!41 7 7 1 7 7 7 1 7 > tool speciRes t#e end e/ector direction

7 7 71 7 7 7 1

88mdlt%olink > script to directly produce a t%o link robott%olink9t%o link :" axis+ ''+ stdD-+slo%'GJ= from Spong+-utc#inson+ 5idyasagar

1"


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@@@T@@@@@@@@@@@ @@@@@@@@@@@@@@@@@@@@@@ @@@@@@@@@@@ U 0 U t#eta U d U a U alp#a U@@@ T@@@@@@@@@@@

@@@@@@@@@@@ @@@@@@@@@@@ @@@@@@@@@@@

U 1U 1U 7U 1U 7UU "U "U 7U 1U 7U@@@ T@@@@@@@@@@@

@@@@@@@@@@@ @@@@@@@@@@@ @@@@@@@@@@@

grav 9 7 base 9 1 7 7 7 tool 9 1 7 7 7

77 1 77 7 1 7 7

6!417 7 17 7 7 1 7

7 7 7 1 7 7 7 188

t%olink!n> get number of links inrobot

ans 9

"

88 links 9t%olink!links links9

t#eta91+d9

t#eta9"+d9

7+

a9

7+

a9

> get link information

1+ alp#a9 7 :'+stdD-=1+ alp#a9 7 :'+stdD-=

88 t%olink!fkine:B7 7= > fkine gives t#e pose :.x.= of t#e robot end@e/ector as a #omogeneous transformation

ans9

1 7 7 "7 1 7 77 7 1 7

7 7 7 1

> t%olink!fkine:Bpi for t#eta 9 B7 7 T


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1&


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88 t%olink!plot:Bpi


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9 7 7 77 7 1 7 7 7 1 7 7

6!41 7 7 1 7 7 7 1 77 7 7 1 7 7 7 1

1.


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88p27!plot:N=

88p27!plot:r=

88p27!plot:s=

88p27!plot:n=


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1


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Code:88 :1= 9 Link:B7 7. 7= 9

t#eta9+ d97+a9 7+ alp#a9

7:'+stdD-=

88 :"= 9 Link:B7 7& 7= 9

t#eta91+d9

7+a9 7+ alp#a9

7:'+stdD-=

t#eta9"+d9

7+a9 .+ alp#a9

7:'+stdD-=

88 :&= 9 Link:B7 7" 7=

9

t#eta91+d9

7+a9 .+ alp#a9

7:'+stdD-=

t#eta9"+d9

7+a9 &+ alp#a9

7:'+stdD-=

t#eta9&+d9

7+a9 "+ alp#a9

7:'+stdD-=

88 S)M 9 SerialLink:+ name+myot=

S)M 9 Tinku :& axis+ '''+ stdD-+slo%'GJ=

T@@@ T@@@@@@@@@@@ @@@@@@@@@@@T@@@@@@@@@@@

@@@@@@@@@@@ T

U 0 U t#eta U d U a U alp#a U

T@@@ T@@@@@@@@@@@ T@@@@@@@@@@@T@@@@@@@@@

@@

@@@@@@@@@@@ T

U 1U 1U 7U .U 7UU "U "U 7U &U 7UU &U &U 7U "U 7U

T@@@ T@@@@@@@@@@@T@@@@@@@@@

@@

T@@@@@@@@@

@@

@@@@@@@@@@

@ Tgrav9

7 base 9 1 7 7 7 tool 9 1 7 77

7 7 1 7 7 7 1 7 7

6!41 7 7 1 77 7 1

77 7 7 1 7 7 7 1

> dr 9 pi 1 9 B7 7 7;dr1 9

7 7 7


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> " 9 B17 "7 &7;dr" 9

7!13. 7!&.61 7!"&2> & 9 B67 67 67;dr& 9

1!3741!374

1!37

4 T&71 9S)M!fkine:1=

T&719

1 7 7 67 1 7 77 7 1 7

7 7 7 1

88 T&7" 9 S)M!fkine:"=

T&7"9

7!777 @7!4227 7 3!&3&

12


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7!4227 7!777

7&!6"22

7 7 1!7777 77 7 7 1!7777

> T&7& 9 S)M!fkine:&= T&7& 9

@7!7777 1!7777 7

@&!7777

@1!777

7 @7!7777 7"!777

77 7 1!7777 77 7 71!7777

S)M!plot:B7 7 7+ %orkspace+ B@ @ @ =

S)M!plot:B17;dr+ "7;dr+ &7;dr+ %orkspace+ B@ @ @ =

S)M!plot:B67;dr+ 67;dr+ 67;dr+ %orkspace+ B@ @ @ =


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Experiment

Aim: !o st"d# t$e %n&erse 'inematics of a panar degree offreedom robot "sing Peter Cor*e toobox

!$eor#:We have shown how to determine the pose of the end-eector giventhe joint coordinates and optional tool and base transforms. A problem of real practical interest is the inverse problem: given thedesired pose of the end -effector ξE what are the required jointcoordinates !or e"ample# if we $now the %artesian pose of an object#what joint coordinates does the robot need in order to reach it &his isthe inverse $inematics problem which is written in functional form as

'n general this function is not unique and for some classes ofmanipulator no closed form solution e"ists# necessitating a numerical solution.

> mdlpuma27> n

n 9

7 7!34. &!1.12 7 7!34. 7

88 T 9 p27!fkine:n= > fkine gives t#e pose :.x.= of t#e robot end@e/ector as a #omogeneous transformation

T9@7!7777 7!7777 1!77777!62& @7!7777 1!7777

@7!7777 @7!171 @1!7777@7!7777 @7!7777 @7!71..

7 77 1!7777

> i 9 p27!ikine2s:T= > ikine2s gives t#e 0oint coordinates corresponding to t#erobot end@e/ector

%it# pose T

i 9@7!77777!34.&!1.12&!1.12 @7!34. @&!1.12

88 i 9p27!ikine2s:T=i 9@7!77777!34.&!1.12&!1.12 @7!34. @&!1.12

88

N

N

9

7 7 7 7 7 7

88 T 9 p27!fkine:N=


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T 91!7777 7 7 7!."1

14


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7 1!7777 7

@7!17

7

7 71!77777!.&1

47 77 1!7777

> i 9 p27!ikine2s:T=i 9

7 1!".6 @&!7.32 @&!1.12 @1!""4 @&!1.12

> N9Bpi i 9 p27!ikine2s:T=OarningQ point notreac#able 8 (nSerialLink!ikine2s at "71

i 9GaG GaG GaG GaG GaG GaG

88 i 9 p27!ikine2s:T+ ru= > gives t#e original set of 0oint angles byspecifying a rig#t #anded conRguration %it# t#e elbo% up

i 9"!2.42 @"!&741 &!1.12 @"!.23& @7!427.

@7!.47 88 p27!plot:i=

88 p27!ikine2s: transl:1+ 7+ 7= = > ,ue to mechanical limits on -oint angles and possible collisions between lins not all eight solutions are ph/sicall/ achievable0 no solution can be achieved

OarningQ point notreac#able 8 (nSerialLink!ikine2s at "71ans 9


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GaG GaG GaG GaG GaG GaG

88 p27!ikine2s: transl:7!+ 7!+ 7!= =

16


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ans 97!666"7!343& @1!4. &!1.12 @1!74" "!1.".

Code:

Orite do%n a Matlab program to solve planar ''' robot inverse kinematicsproblem #aving t#e lengt# parameters L19.m+ L"9&m+ L&9"m!

$or t#e follo%ing relative pose

:a= 7 T- 9 B1 7 7 6 7 1 7 7 7 7 1 7 7 7 7 1

:b= 7 T- 9 B7! @7!422 7 3!&3& 7!422 7!2 7 &!6"22 7 7 1 7 7 7 7 1

:c= 7 T- 9 B7 1 7 @& @1 7 7 " 7 7 1 7 7 7 7 1

:d= 7 T- 9 B7!422 7! 7 @&!1". @7! 7!422 7 6!123. 7 7 1 7 7 7 7 1

:a= T-79

1 7 7 67 1 7 77 7 1 77 7 7 1

dr 9pi


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7 27 7 1!7777 77 77 1!7777

> 1 9 anadi!ikine:T-1+ B7 7 7+ B1 1 7 7 7 1=

matrix not ort#onormal rotation matrix

"7


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T-1:1Q&+1Q&=;T-1:1Q&+1Q&= > T#e above can be proved as

ans 9 > for ort#onormal matrix t#e result s#ould be identitymatrix

1!7777 @7!7422 7@7!7422 1!1177 7

7 7 1!7777:c= T-" 9 B7 1 7 @& @1 7 7 " 77 1 7 7 7 7 1 T-" 9

7 1 7 @&@1 7 7 "7 7 1 77 7 7 1

88 S" 9 anadi!ikine:T-"+ B7 7 7C+ B1 17 7 7 1C=

"

9 anadi!ikine:T-"+ B67 @67 67C;dr+B1 1 7 7

71=

"9 " 9

1!374 1!374 1!374 "!434 @1!374 @"!434

88anadi!plot:"= 88anadi!plot:"=

T-& 9

7!4227 7!777 7 @&!1".@7!777 7!4227 7 6!123.

7 7 1!7777 77 77 1!7777

> & 9 anadi!ikine:T-&+ B7 7 7;dr+ B1 17 7 7 1= & 9

1!24" 1!22.6"!.76anadi!fkine:&=ans 9

7!4227 7!777 7 @1!262@7!777 7!4227 7 "!&"6

7 7 1!7777 77 7 7 1!7777

S&

9 anadi!ikine:T-&+ B67 @67 67;dr+ B1 1 77 7 1= >

for di/erent initial estimate


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S& 9

"1


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1!7e7. ;@"!23.22!.6&" @&!3741

88anadi!fkine:&=

ans 97!4326 7!.474 7 @1!21""

@7!.474 7!4326 7 "!&33"

7 71!777

7 77 7 7 1!7777

S& 9 anadi!ikine:'+ B7 7 7C+ B" 7 77 7 1C=

>for di/erentmask

& 9@"!72 @7!673 "!..6.

88anadi!fkine:&=

ans 97!42277!777 7@&!1".

@7!7777!4227

7@!7"&4

7 7 1!7777 77 7 7 1!7777

88anadi!plot:&=

T#e di/erent result for t#e same post7 T-is due to di/erent usage of (nitial

estimate :X7= and mask

:M=!


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""


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Experiment +

Aim: !o perform for,ard *inematics of D- panar robot/ Stanfordmanip"ator / p"ma robot for t$e gi&en D0 parameters "sing roboana#er and cac"ate t$e end eectors position for a gi&en set of 3oint &a"es "sing matab and &erif# t$e res"t ,it$ robo ana#er

o"tp"t.

!$eor#:

$or%ard )inematicsQ To determine t#e position and orientation of t#e end@e/ector given t#e values for t#e 0oint variables of t#e robot!

,sing 'oboanalyserQ 'oboAnalyNer is a &D Model ased 'obotics LearningSystem developed using *penT) and 5isual CY!

T#e aboveRgures#ave beensuitably

labeled for proper description! T#e ,( may be explained as follo%sQ


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1! (n t#is block+ one can select t#e D*$ and type of robot from tabs providedon t#e far left! T#e center table s#o%s t#e D- parameter for t#e robot!

T#ese can be c#anges as per t#e reuirement!"! -ere . tabs are present! *ne can see a D- paremeter animation for t#e

robot in Z5isualiNe D-[ tab! ZLink ConRg[ and ZJJ ConRg[ gives t#etransformation matrix for a particular link and for base to end@e/ectorrespectively!

&! *nce t#e robot is conRgured+ press t#e Z$)in[ button to completer t#efor%ard kinematic analysis! T#en press t#e play button to vie% t#etra0ectory of robot %it# initial and Rnal position as speciRed in block 1!*ne can also c#ange t#etime duration and no! of steps taken by robot forreac#ing from initial to Rnal position!

.! -ere one can see t#e actual robot+ coordinate axis %!r!t eac# link+ and t#etra0ectory plotted after #itting t#e play button!

! T#is screen is obtained by #itting t#e Z\rap#[ tab on top of t#e screen!-ere one can see t#e variation in position for eac# link separately %it#time and t#e 0oint parameters suc# as 0oint value+ velocity+ accelerationetc!

2! T#is block s#o%s t#e grap# for selected parameter in block !

D- panar robot88 Link

ans 9t#eta9+ d9 7+ a9 7+ alp#a9 7 :'+stdD-=> L:1= 9 Link:B7 7 7!& 7=> L:"= 9 Link:B7 7 7!" pi


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grav9

7 base 9 1 7 7 7 tool 9 17 7 7

7 7 1 7 7 7 1 7 76!41 7 7 1 7 7 7 1 7

7 7 7 1 7 7 7 1

88 D'9pi


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88 S19B 7 677C; D'

1 9

71!37

4 7

88 T&719fkine:t#reelink+1= T&719

7!7777@7!77771!7777

7!&777

1!7777 7!7777@

7!77777!&77

71!777

7 7!7777 77 7 7 1!7777

Stanford manip"ator

88 Link

ans 9t#eta9+ d9 7+ a9 7+ alp# a9 7 :'+stdD-=> L:1= 9 Link:B7 7!32" 7 @pi L:&= 9 Link:B@pi L:= 9 Link:B7 7 7 @pi


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d9 alp#a9 :'+stdD-=t#eta9"+d9 7!&6&.+ a9

7+alp#a9

@1!31:'+stdD-=

t#eta9 @1!31+d9&+ a9

7+alp#a9

7:P+stdD-=

"


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t#eta9.+d9

7!""24+a9

7+ alp#a9 @1!31:'+stdD-=

t#eta9+d9 7+ a9

7+alp#a9

@1!31:'+stdD-=

t#eta92+d9

7!.&14+a9

7+alp#a9

7:'+stdD-=

88 Stanlink 9 SerialLink:L+ name+ Stan link=

Stanlink 9Stan link :2 axis+ ''P'''+ stdD-+ slo%'GJ=

T

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U 1U S1U 7!32"U 7U @1!31U

U "U S"U7!&6&.

U 7U @1!31UU &U @1!31U &U 7U 7U

U .U S.U7!""24

U 7U @1!31UU U SU 7U 7U @1!31U

U 2U S2U7!.&14

U 7U 7U

T

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grav9

7 base 9 1 7 77 tool 9 1 7 7 7

7 7 1 7 7 7 1 7 7

6!41 7 7 1 7 7 7 1 77 7 7 1 7 7 7 1

88 19B 7 @pi


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"2


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P"ma robot88 L:1= 9 Link:B7 7!33."1 7 @pi


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UT@@@ T

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U 1U 1U 7!33."U 7U @1!31U

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T@@@

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7 base 9 1 77 7 tool 9

1 7 77

77 1 7 7

7 1 7

76!41 7 7 1 7 7 7 1 7

7 7 7 1 7 7 7 188 19B 7 7 7 77 71 9

7 7 7 7 7 788

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T2719

1!7777 7 7 7!"22

71!777

7 7 7!1&63

7 71!777

7 1!1772

7 7 7 1!7777


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"

4


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Experiment 4 To perform inverse kinematics of & D*$ planar robot of t#e

given D- parametersAimof$e experiment:

usingroboanalyNer! Derive t#e

analytical expression for t#e inverse kinematics problem of a " D*$ planar robot and

verify t#e solution from Matlab program!

%n&erse 'inematics (%'in) consists of determination of t#e 0oint variablescorresponding to a given end]e/ectors orientation and position! T#e solution tot#is problem is of fundamental importance in order to transform t#e motionspeciRcations assigned to t#e end]e/ector in t#e operational space into t#ecorresponding 0oint space motions! T#ere may be multiple or no results possiblefor a given end]e/ector position and orientation!

23+T425 2' 4645To select a robot and view the solutions of its 4nverse 6inematics! the following are the steps

1. 7lic on 46in button. 4t shows a separate window ('igure 18*". elect a 9obot#. :nter 4nput parameters;. 7lic on 46in buttoning 46in window?. elect an/ of the obtained solution as initial and final solution@. 7lic on 26. This step replaces the initial and final -oint values in ,A Barameter table

Code:

T#e solution may be obtained from Matlab using trigonometric analysis ordirectly from 'oboAnalyNer!

(a) D- panar robot


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"6


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a19" a"9"a&91

> Gon@Nero constant D-parameters

p#i9pi (nput

%x 9 px @ a&;cos:p#i=%y 9 py@a&;sin:p#i= del9%x;%x

%y;%yc" 9 :del@a1;a1@a";a"=<:";a1;a"=

> Calculation fort#eta"

s" 9 sSrt:1@c";c"=t#"19atan":s"+c"= t#""9atan":@s"+c"=

>Calculation for t#eta1

s119 ::a1a";cos:t#"1==;%y@a";sin:t#"1=;%x=


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&7


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(b) Artic"ated

> a"91 a&91

88px91py97

pN97

88delxy9px;pxpy;py del9delxypN;pN

> t#119atan":py+px=

> t#1"9piatan":py+px=

> c&9:del@a";a"@a&;a&= t#&19atan":s&+c&= t#&"9atan":@s&+c&=

> s"19:@:a"a&;cos:t#&1==;pN@a&;s&;delxy= c"19::a"a&;cos:t#&1==;delxya&;s&;pN= s""9:@:a"a&;cos:t#&1==;pNa&;s&;delxy= c""9::a"a&;cos:t#&1==;delxy@a&;s&;pN= t#"19atan":s"1+c"1=t#""9atan":s""+c""=

> t#"&9pi@t#"1 t#".9pi@t#""

> r"d9147 t#11d9t#11;r"d

> t#1"d9t#1";r"d


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> t#"1d9t#"1;r"d

> t#""d9t#"";r"d

> t#"&d9t#"&;r"d

&1


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> t#".d9t#".;r"d

> t#&1d9t#&1;r"d

> t#&"d9t#&";r"

d

> t#11d9t#11;r"d

oboAna#er-"tp"t


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&"


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Experiment 5

Aim: To study velocity kinematics using Peter Corke tool box!

!$eor#:

6eocit# 'inematics: 5elocity )inematics relates linear and angular

velocities oft#e end@e/ector to t#e 0oint velocities!

88mdlpuma2788 T7 9p27!fkine:n= > obtain transformation matrix

T7 9

@7!77777!7777 1!7777 7!62&

@7!77771!7777 @7!7777@7!171

@1!7777@7!7777 @7!7777@7!71..

7 7 71!777

788d 9 1e@2 > give slig#t disturbance88Tp 9 p27!fkine:n Bd 7 77 7 7=

> obtain ne% transformationmatrix

Tp 9

@7!7777

@7!7777 1!7777 7!62&

@7!7777 1!7777 7!7777 @7!177

@1!7777@7!7777 @7!7777@7!71..

7 7 71!777

788 dTd1 9 :Tp @

T7= < d > obtain t#e di/erentialdTd19

7@

1!7777 @7!7777 7!177

@7!7777

@7!7777 1!7777 7!62&

7 7 7 77 7 7 788 d'd1 9dTd1:1Q&+1Q&=

> get t#e &x& di/erentialrotation matrix

d'd19

7@

1!7777 @7!7777@7!7777 @ 1!7777


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7!7777

7 7 788 ' 9

T7:1Q&+ 1Q&=> obtain rotational matrix from

T7

'9

@7!7777 7!7777

1!7777 @7!7777

1!7777 @7!7777

&&


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@1!7777 @7!7777 @7!7777

88 S 9 d'd1 ; ' > obtain t#e ske% symmetric matrix

S9

@7!7777 @1!7777 7!7777

1!7777 @7!77777!77777 7 7

88vex:S=

> get vector from symmetricmatrix

ans9@7!7777

7!7777

1!777788 ^ 9p27!0acob7:n=

> get 0acobian %!r!t! %orldcoordinates

^ 9

7!1717!71.. 7!&163 7 7 7

7!62&7!7777 7!7777 7 7 7

7 7!62&7!"617 7 7 7

7 7!77777!7777 7!3731

7!7777 1!7777

7!7777 @1!7777 @1!7777 @7!7777 @1!7777 @7!7777

1!77777!7777

7!7777

@7!3731 7!7777@7!7777

88T 9 transl:1+ 7+7=;troty:pi get ne% 0acobian matrix %!r!t! ne% frame

^ 9

7!7777 7@1!7777 7 1!7777 77 1!7777 7 7 71!7777

1!7777 77!7777 7

@7!7777 7

7 7 7 7!7777 7@

1!7777


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7 7 7 71!7777 77 7 7 1!7777 77!7777

88v 9 ^;B1 7 7 7 7 7

88v

ans 9

7!7777 7 1!7777 7 7 7

&.


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88p27!0acobn:Sn=

> get 0acobian matrix %!r!t! end@e/ector

ans9

@7!7777 @7!62&@7!"617 7 7 7

7!62& 7!7777 7!7777 7 7 77!177 7!71.. 7!&163 7 7 7

@1!7777 7 77!373

1 7 7

@7!7777@

1!7777@

1!7777@

7!7777

@1!7777 7

@7!7777 7!77777!7777 7!3731

7!7777

1!7777

88p27!0acob7:n+eul=

> get 0acobian matrix for Jularvalues

ans9

7!171 7!71.. 7!&163 7 7 77!62& 7!7777 7!7777 7 7 7

7 7!62&7!"617 7 7 7

1!7777 7!7777 7!7777@7!3731 7!7777

7!7777

7!7777@

1!7777@

1!7777@

7!7777

@1!7777 7

@7!7777 7!77777!7777 7!3731

7!7777

1!7777

To study inverse kinematics and develop a MATLA program using petercoorke toolbox to calculate a 0acobianmatrix for planar &' robot+ given t#e

robots lengt# l19.m+ l"9&m+ l&9"m and initial 0oint angles _ :_1+_"+_&=9B17o+

"7o+&7o!

88L:1= 9 Link:B7 7 . 7 =

88L:"= 9 Link:B7 7 & 7 =

88L:&= 9 Link:B7 7 " 7 =

L 9

t#eta91+d9 7+ a9 .+alp#a9

7

:'+stdD-=

t#eta9"+d9 7+ a9

&+alp#a9

7:'+stdD-=

t#eta9&+d9 7+ a9

"+alp#a9

7:'+stdD-=

88t%olink0acob 9 SerialLink:L+ name+t%olink0acob= t%olink0acob 9t%olink0acob :& axis+ '''+ stdD-+ slo%'GJ=


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T

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U 1U 1U 7U .U 7U

U "U "U 7U &U 7UU &U &U 7U "U 7U

T

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@@@@@@@@@@@

grav9

7 base 9 1 77 7

tool 9 1 7 77

7 7 1 7 7 7 1 7 76!41 7 7 1 7 7 7 1 7

7 7 71 7 7 7 1

88dr9 pi


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dr 97!713

88 7 9 B17 "7 &7;dr7 97!13. 7!&.61

7!"&2

880acobian 9 0acob7:t%olink0acob+ 7= 0acobian 9

@&!6"22 @&!"&"1 @1!3&"13!&3

&&!641

1!7777

7 7 77 7 77 7 7

1!7777

1!7777

1!7777


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&2


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Experiment 7

AimQ To study for%ard dynamics of a &D*$ of &' robot!

!$eor#:

DynamicsQ'obot dynamics is concerned %it# t#e relations#ip bet%een

t#e forces acting on a robot mec#anism and t#e accelerations t#eyproduce! $or%ard dynamics %orks out t#e accelerations given t#e forces!

88mdlpuma27

X 9 p27!rne:n+ N+N=

>0oint torue reuired for t#e robot ' to ac#ieve t#especiRed 0oint

position X+ velocity XD andacceleration XDD!

X 9@7!777

7

&1!2&6

62!7&1 7!77777!7"4& 7X 9 p27!rne:n+ N+ N+ B7 76!41= > B7 7 6!41 gives t#e gravitational vector

X 9@7!7777

&1!2&662!7&1 7!77777!7"4& 7

B+ 1+ " 9 0tra0:N+r+ 17=

> Compute a 0oint space tra0ectory bet%eent%o points

p27!links:1=!dyn


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&3


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p27!gravity

> Default gravity assumed bypuma robot

ans 9

7 76!4177

88p27!gravload:n=> TorSue reSuired to maintain t#e robot inposition

ans 9

@7!7777 &1!2&66 2!7&17!7777 7!7"4&

p27!gravity9 p27!gravity 'obot in Ws position

X 9 @7!7777 .2!7726 4!33"" 7!7777 7!7"4& 7

88 X 9 p27!gravload:r=

X 9

7 @7!33" 7!".46 7 7 7

8. To study for%ard dynamics of & D*$ &' robot and develop a MATLAprogram using peter coorke toolbox to calculate t#e for%ard dynamics for

planar &' robot+ given t#e robots parameters as L19.m+ L"9&m+ L&9"m and

m19"7)g+ m"91)g+ m&917)g and 97!)gm"+ 97!")gm"+

97!1)gm"!(gnore gravity by assuming t#at t#e gravity acts in a direction normal to t#eplane of motion! ,sing Peter Coorke toolbox and Matlab commands solve t#efor%ard dynamics problem :i!e! %it# t#eavailable driving 0oint torues an 0oint angles and initial 0oint rates=! Performt#is simulation for .

sec! 9 E + + F 9 E + + F


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&4


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9 E + + F 9 ` + +

!

F 9> Link1

+ +%it# a9.m+ m9"7kg+ (97!kgm

"L:1=

9 Link:B7 7 . 7 7 "7 "7 7 7 7

7!

7 7 7 7 1 7 7 7=

9 E + +

L:"

=

9 Link:B7 7 & 7 7 1 1! 7 7 7 7 7!" 7 7

7 7 1 7 7 7C=

> Link" %it# a9&m+ m91kg+

(97!"kgm"

L:&=

9 Link:B7 7 " 7 7 17 17 7 7 7

7!1 7 7 7 7 17 77=

> Link& %it# a9"m+ m917kg+

(97!1kgm"

L:1=!md#91 > ,se modiRed D- parameters

L:"=!md#91

L:&=!md#91

t%olinksau 9 SerialLink:L+ name+ t%o linksau=

t%olinksau 9

t%olinksau :& axis+ '''+ stdD-+ slo%'GJ=

T

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U 1U 1U 7U .U 7UU "U "U 7U &U 7UU &U &U 7U "U 7U

T

@@@T

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@@@@@@@@@@@

@@@@@@@@@@@

@@@@@@@@@@@

grav9 7 base 9 1 77 7 tool 9 1 7 777 7 1 7 7 7 1 7 7

6!41 7 7 1 7 7 7 1 77 7 7

1 7 7 7 1

t797 tf9. G9.7 dt9:tf@t7=


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Experiment 8

AIM : To study Peter CorkeComputer Vision toolbox

Different Functions and Explanation:

//Spectral Representation of Light//

//define a range of wavelengths//

>>lambda = [300:10:1000]*1e-9; //in this case from 300 to 1 000 nm, and then compute the blackbody spectra// >>forT=1000:1000:6000>>plot( lambda*1e9, blackbody(lambda, T)); hold all >>end

as shown in Fig. 8.1a.

//The filament of tungsten lamp has a temperature of 2 600 K and glows white hot. The Sun has a surfacetemperature of 6 500 K. The spectra of these sourcesare compared in Fig. 8.1b.//

>>lamp = blackbody(lambda, 2600);>>sun = blackbody(lambda, 6500);>>plot(lambda*1e9, [lamp/max(lamp) sun/max(sun)])

//The Sun’s spectrum at ground level on the Earth has been measured and tabulated//

>>sun_ground = loadspectrum(lambda, 'solar.dat');>>plot(lambda*1e9, sun_ground)and is shown in Fig. 8.3a.

.1


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AIM: Absorption

> [A, lambda] = loadspectrum([400:10:700]*1e-9,

'water.dat'); //A is the absorption coefficient// > d = 5; // d is the path length//

> T = 10. (̂-A*d);// Transmission T is the inverse of absorption// >>plot(lambda*1e9, T);

which is plotted in Fig. 8.3b.

Reflection

//the reflectivity of a red housebrick// >> [R, lambda] = loadspectrum([100:10:10000]*1e-9, 'redbrick.dat');>>plot(lambda*1e6, R);

."


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which is plotted in Fig. 8.4. We see that it reflects red colors more than blue.

//The illuminance of the Sun in the visible region// >>lambda= [400:10:700]*1e-9; % visible spectrum >> E =loadspectrum(lambda, 'solar.dat');

//at ground level. The reflectivity of the brick is// > R = loadspectrum(lambda, 'redbrick.dat'); //andthe light reflected from the brick is// > L = E .* R;>>plot(lambda*1e9, L);

//which is shown in Fig. 8.5. It is this spectrum that is interpreted by our eyes as the color red.//

Color

//The luminosity function is provided by the Toolbox//>>human = luminos(lambda);>>plot(lambda*1e9, human) //andis shown in Fig. 8.6a. //

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//The spectral responses of the cones can be loaded and shown in fig 8.7.b// >>cones =loadspectrum(lambda, 'cones.dat');>>plot(lambda*1e9, cones)

Reproducing Colors

// cmfrgb function return the color matching functions//

>>lambda = [400:10:700]*1e-9;

>>cmf = cmfrgb(lambda);>>plot(lambda*1e9, cmf);

.

.


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>>orange = cmfrgb(600e-9)// create the sensation of light at 600 nm (orange)// orange =

2.8717 0.3007 -0.0043

>>green = cmfrgb(500e-9)green =-0.2950 0.4906 0.1075

>> w = -green(1);>>white = [w ww ];>>feasible_green = green + whitefeasible_green =0 0.7856 0.4025

>>whitewhite =0.2950 0.2950 0.2950

//The Toolbox function cmfrgbcan also compute the CIE tristimulus for an arbitrary spectralresponse. //

>>RGB_brick = cmfrgb(lambda, L)RGB_brick =0.6137 0.1416 0.0374Chromaticity Space

// The Toolbox function lambda2rg computes the color matching function //

>> [r,g] = lambda2rg( [400:700]*1e-9 );>>plot(r, g)>>rg_addticks

>>primaries = cmfrgb( [700, 546.1, 435.8]*1e-9 );>>plot(primaries(:,1), primaries(:,2), 'd')

.


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//chromaticity of the spectral green color//

>>green_cc = lambda2rg(500e-9); green_cc =

-0.9733 1.6187>>plot2(green_cc, 's')

>>white_cc = tristim2cc([1 1 1])//of white color//white_cc =0.3333 0.3333>>plot2(white_cc, '*')

Color Names

>>colorname('?burnt') ans='burntsienna' 'burntumber'

>>colorname('burntsienna') ans=0.5412 0.2118 0.0588

>>bs = colorname('burntsienna', 'xy') bs =

0.5258 0.3840

>>colorname('chocolate', 'xy') ans=0.5092 0.4026

>>colorname([0.54 0.20 0.06]) ans=burntsienna

>>colorname(bs, 'xy')ans ='burntsienna'

Other Color Spaces

//The function colorspacecan be used to perform conversions between different color spaces. //

>>colorspace('RGB->HSV', [1, 0, 0]) ans =

.

2


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0 1 1>>colorspace('RGB->HSV', [0, 1, 0]) ans =

120 1 1>>colorspace('RGB->HSV', [0, 0, 1]) ans =

240 1 1 // the saturation is 1, the colors are pure, and the intensity is 1//

>>colorspace('RGB->HSV', [0, 0.5, 0]) ans =

120.0000 1.0000 0.5000 // intensity drops but hue and saturation are unchanged//

// Image Formation //

cam = CentralCamera('focal', 0.015); // creatinga model of a central-perspective camera // >> P = [0.3, 0.4, 3.0]';>>cam.project(P);

ans =0.00150.0020

cam.project(P, 'Tcam', transl(-0.5, 0, 0) ) // moving camera 0.5 m to the left // ans =

0.00400.0020

cam = CentralCamera('focal', 0.015, 'pixel', 10e-6, ... // displa/s the parameters of the imaging modelCC

'resolution', [1280 1024], 'centre', [640 512], 'name', 'mycamera') name:mycamera [central-perspective]focal length: 0.015pixel size: (1e-05, 1e-05)principal pt: (640, 512) numberpixels: 1280 x 1024

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T:1 0 0 00 1 0 00 0 1 00 0 0 1>>cam.project(P);ans =

790712

>>cam.Cans =

1.0e+03 *

1.5000 0 0.6400 0

0 1.5000 0.5120 0

0 0 0.0010 0

>>cam.fov() * 180/pi // field of view // ans =

46.2127 37.6930

// we create a 3×

3 grid of points in the xy-plane with overall side length 0.2 m and centred at (0, 0, 1) ,returns 3*9 matrix //

>> P = mkgrid(3, 0.2, 'T', transl(0, 0, 1.0));>>P(:,1:4);ans =

-0.1000 -0.1000 -0.1000 0

-0.1000 0 0.1000 -0.1000

1.0000 1.0000 1.0000 1.0000

>>cam.project(P) // The image plane coordinates of the vertices are //

ans =

490 490 490 640 640 640 790 790 790

362 512 662 362 512 662 362 512 662

>>cam.plot(P) h=

173.0022

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>>Tcam = transl(-1,0,0.5)*troty(0.9); // oblique view of the plane // >>cam.plot(P,'Tcam', Tcam)

>>cam.project([1 0 0 0]', 'Tcam', Tcam) ans =

1.0e+03 *

1.8303

0.5120>> p = cam.plot(P, 'Tcam', Tcam)

>>p(:,1:4) // oblique viewing case the image plane coordinates //

ans =

887.7638 887.7638 887.7638 955.2451

364.3330 512.0000 659.6670 374.9050

>>cube = mkcube(0.2, 'T', transl([0, 0, 1]) ); // projection of cube //

>>cam.plot(cube);

>> [X,Y,Z] = mkcube(0.2, 'T', transl([0, 0, 1.0]), 'edge');

>>cam.mesh(X, Y, Z)>>cam.T = transl(-1,0,0.5)*troty(0.8); // oblique view // >>cam.mesh(X,Y, Z, 'Tcam', Tcam);

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/ll //// Camera Calibration // // lHomogeneous Transformation Approach //

>> P = mkcube(0.2);>>T_unknown = transl(0.1, 0.2, 1.5) * rpy2tr(0.1, 0.2, 0.3);>>cam = CentralCamera('focal', 0.015, ...'pixel', 10e-6, 'resolution', [1280 1024], 'centre', [512 512], ...'noise', 0.05);

> p = cam.project(P, 'Tobj', T_unknown)> C = camcald(P, p)maxm residual 0.067393 pixels.

C =883.1620 -240.0720 531.4419 612.0432259.3786 994.1921234.8182 712.0180

-0.1043 0.0985 0.6494 1.0000

// lDecomposing the Camera Calibration Matrix //est = invcamcal(C) est=name: invcamcal [central-perspective] focallength: 1504pixel size: (1, 0.9985)principal pt: (518.6, 505)

Tcam:0.93695 -0.29037 0.19446 0.083390.31233 0.94539 -0.093208 -0.39143-0.15677 0.14807 0.97647 -1.46710 0 0 1

>>est.f/est.rho(1) // ratio of estimated parameters of camera // ans =

1.5044e+03

>>cam.f/cam.rho(2) // ratio of true parameters of camera //

ans =

1.500e+03

>>T_unknown*est.T // relative pose between the true and estimated camera pose//

ans =

0.7557 -0.5163 0.4031 -0.0000

0.6037 0.7877 -0.1227 -0.0001

-0.2541 0.3361 0.9069 -0.0041

0 0 0 1.0000

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>>plot_sphere(P, 0.03, 'r')>>plot_frame(eye(4,4), 'frame', 'T', 'color', 'b', 'length', 0.3)>>est.plot_camera()

// Pose Estimation //

>>cam = CentralCamera('focal', 0.015, 'pixel', 10e-6, ...'resolution', [1280 1024], 'centre', [640 512]); // create acalibrated camera with known parameters//

>> P = mkcube(0.2); // cube determination // >>T_unknown = transl(0,0,2)*trotx(0.1)*troty(0.2) // pose estimation w.r.t camera// T_unknown =

0.9801 0 0.1987 0

0.0198 0.9950 -0.0978 0

-0.1977 0.0998 0.9752 2.0000

0 0 0 1.0000

>> p = cam.project(P, 'Tobj', T_unknown);

>>T_est = cam.estpose(P, p) // estimate relative pose of the object //

T_est =

0.9801 0.0000 0.1987 -0.0000

0.0198 0.9950 -0.0978 -0.0000

-0.1977 0.0998 0.9752 2.0000

0 0 0 1.0000

// lNon-Perspective Imaging Models //

>>cam = FishEyeCamera('name', 'fisheye', ...'projection', 'equiangular', ...'pixel', 10e-6, ...'resolution', [1280 1024]); //creating fishery camera // >> [X,Y,Z] = mkcube(0.2, 'centre', [0.2, 0, 0.3], 'edge'); // creating edge based model //>>cam.mesh(X, Y, Z)

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l// // lCatadioptric Camera //>>cam = CatadioptricCamera('name', 'panocam', ...'projection', 'equiangular', ...maxangle',pi/4, ... 'pixel', 10e-6,... 'resolution', [1280 1024])>> [X,Y,Z] = mkcube(1, 'centre', [1, 1, 0.8], 'edge');>>cam.mesh(X, Y, Z)

// spherical camera //

>>cam = SphericalCamera('name', 'spherical')>> [X,Y,Z] = mkcube(1, 'centre', [2, 3, 1], 'edge');>>cam.mesh(X, Y, Z)

// lMapping Wide-Angle Images to the Sphere //>>fisheye = iread('fisheye_target.png', 'double', 'grey');> u0 = 528.1214; v0 = 384.0784; // camera calibration parameters // > l=2.7899;> m=996.4617;

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> [Ui,Vi] = imeshgrid(fisheye); // domain of input image // > n = 500;>>theta_range = (0:n)/n*pi/2 + pi/2;>>phi_range = (-n:2:n)/n*pi;> [Phi,Theta] = meshgrid(phi_range, theta_range); // n is no of grid cells // > r = (l+m)*sin(Theta) ./ (l+cos(Theta));>> U = r.*cos(Phi) + u0; // Cartesian coordinates in the input image // >> V =r.*sin(Phi) + v0;>>spherical = interp2(Ui, Vi, fisheye, U, V); // applying wrap //>>idisp(spherical)

// lSynthetic Perspective Images //

>> W = 1000;

>> m = W / 2 / tan(45/2*pi/180) // 45 fied of view // m =

1.2071e+03> l = 0;> u0 = W/2; v0 = W/2;>> [Uo,Vo] = meshgrid(0:W-1, 0:W-1); // The domain of the output image // >> r =sqrt((Uo-u0).̂ 2 + (Vo-v0).^2); // polar coordinates // >>phi = atan2((Vo-v0), (Uo-u0)); >>Phi_o =phi; // spherical coordinates // >>Theta_o = pi - acos( ((l+m)*sqrt(r.^2*(1-l^2) + (l+m)^2) -l*r.^2) ./ (r.^2 + (l+m)^2) );>>idisp(perspective)>>spherical = sphere_rotate(spherical, troty(1.2)*trotz(-1.5));

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1 Obtaing an image from a file

The toolbox consist of iread function to read an image

>>street = iread('street.png'); % Which returns a matrix

>>about(street) % Gives details related to street

street [uint8] : 851x1280 (1089280 bytes)

>>street(200,300) % The pixel at image coordinate (300,200)

ans =

42

>>street_d = idouble(street); % double precision number in the range [0, 1].>>about(street_d)>>street_d [double] : 851x1280 (8714240 bytes)

>>street_d = iread('street.png', 'double'); % specify this as an option whileloading>>idisp(street); % to display an image is idisp

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>>flowers = iread('flowers8.png');>>about(flowers)flowers [uint8] : 426x640x3 (817920 bytes)

>>pix = flowers(276,318,:) %pixel at (318, 276)ans(:,:,1) =57 ans(:,:,2)= 91ans(:,:,3) =198

% The pixel value is


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>>idisp( flowers(:,:,1) )

2 Images from an Attached Camera

Most laptop computers today have a builtin camera for video conferencing. For computerswithout a builtin camera an external camera can be easily attached via a USB. The meansof accessing a camera is operating system specificand the Toolbox provides a simple interface to a camera for MacOS, Linux and Windows.

% A list of all attached cameras and their capability can be obtained by>>VideoCamera('?')% We open a particular camera>>cam = VideoCamera('name')% which returns an instance of a VideoCamera object that is a subclass of the%ImageSource class.% The dimensions of the image returned by the camera are given by the size method

>>cam.size()% an image is obtained using the grab method

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>>im = cam.grab();% which waits until the next frame becomes available.

3 Images from a Movie File

The Toolbox supports reading frames from a movie file stored in any of the popular

formats such as AVI, MPEG and MPEG4. For example we can open a movie file>>cam = Movie('traffic_sequence.mpg'); 720x 576 @ 2.999970e+01 fps350 frames

%This movie has 350 frames and was captured at 30 frames per second.

%The size of each frame within the movie is>>cam.size()ans =720 576% and the next frame is read from the movie file by >>im= cam.grab();>>about(im)

im [uint8] : 576x720x3 (1244160 bytes)% which is a 720 × 576 color image.

% With these few primitives we can write a very simple movie player 1 while12 im = cam.grab;3 if isempty(im), break; end4 image(im)5 end%where the test at line 3 is to detect the end of file, in which case grab returns an emptymatrix.

4 Images from Code

% The Toolbox function testpattern generates simple images with a variety of patterns

including lines, grids of dots or squares, intensity ramps and intensity sinusoids.

>>im = testpattern('rampx', 256, 2);>>im = testpattern('siny', 256, 2);>>im = testpattern('squares', 256, 50, 25);

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>>im = testpattern('dots', 256, 256, 100);

We can also construct an image from simple graphical primitives. First we create a

blank canvascontaining all black pixels (pixel value of zero)

>>canvas = zeros(200, 200);

% then we create two squares> sq1 = 0.5 * ones(40, 40);> sq2 = 0.9 * ones(20, 20);The first has pixel values of 0.5 (medium grey) and is 40× 40. The second is smaller(just 20× 20) but brighter with pixel values of 0.9. Now we can paste these onto thecanvas>>canvas = ipaste(canvas, sq1, [20, 40]);>>canvas = ipaste(canvas, sq2, [60, 120]);>>circle = 0.6 * kcircle(30);%of radius 30 pixels with a grey value of 0.6.>>size(circle)ans =61 61

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>>canvas = ipaste(canvas, circle, [100, 30]); %Finally, wedraw a line segment onto our canvas >>canvas =iline( canvas, [30, 40], [150, 190], 0.8);%which extends from (30, 40) to (150, 190) and its pixels are all set to 0.8.>>idisp(canvas)

5 Monadic Operations

Monadic image-processing operations result in an image of the same size W × H as the

input image, and each output pixel is a function of the corresponding input pixel.

Since an image is represented by a matrix any MATLAB® element -wise matrix functionor operator can be applied, for example scalar multiplication or addition, or functions suchabs or sqrt.The datatype of each pixel can be changed, for example from uint8 (integer pixels

in the range [0, 255]) to double precision values in the range [0, 1] >>imd =idouble(im);>>im = iint(imd);

A color image has 3-dimensions which we can also consider as a 2-dimensional imagewhere each pixel value is a 3-vector.

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A monadic operation can convert a color image to a greyscale image where each output

pixel value is a scalar representing the luminance of the corresponding input pixel

>>grey = imono(flowers);The inverse operation is>>color = icolor(grey);

% the histogram of the street scene is computed and displayed by>>ihist( street )

>> [n,v] = ihist(street);

%where the elements of n are the number of times pixels occur with the value of %thecorresponding element of v.

>>ihist(street, 'cdf');

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%The Toolbox function peak will automatically find the peaks>>peak(n, v)'ans =8 40 43 51 197 17 60 26 75 21388 92 218 176 153 108 111 147 113 119121 126 130 138 230 244

We can identify the pixels in

this peak by a logical monadic operation>>shadows = (street >= 30) & (street>idisp(shadows)

>>im = istretch(street);% ensures that pixel values span the full range_ which is either [0, 1] or [0, 255]

%A more sophisticated version is histogram normalization or histogramequalization>>im = inormhist(street);

%Such images can be gamma decoded by a non-linear monadic operation >>im =igamma(street, 1/0.45);which raises each pixel to the specified power, or >>im= igamma(street, 'sRGB');>>idisp( street/64 )

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2"


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Experiment 9

A%: !o st"d# -b3ect trac*ing "sing 'aman ;ter "sing matab.

!$eor#Q

T#e )alman Rlter keeps track of t#ee estimated state of t#e system and t#e

variance or uncertainty of t#e estimate! T#e estimate is updated using a state transition model and measurements! denotes t#e estimate of t#e syste ms state at time step k before t#e

k@t# measurem ent Nk #as been

taken into account is t#e corresponding uncertainty!

T#e )alman Rlter+ also kno%n as linnear uadratic estimation :LXJ=+ is an algorit#mt#a t uses a series of measurements observed over time+ containing noise :randomvariations= and ot#e r inaccuracies+ and produces estimates of unkno% n variablest#at tend to be more precise t#an t#o se based on a single measurement alone!More f ormally+ t#e )alman Rlter operates recursively on streams of noisy input datato produce a statistically optimal estimate of t#e underlying system state!

T#e algorit#m %orks in a t%o@st ep process! (n t#e prediction step+ t#e )almanRlter produces estimates of t#e current state variables+ along %it# t#eiruncertainties! *nce t#e outcome of t#e next measurement :necessarilycorrupted %it# some amount of error+ including ra ndom noise= is observed+t#ese estimates are updated using a %eig#ted average+ %it# more %eig#t beinggiven to estimates %it# #ig#er certainty! ecause of t#e algorit#ms recursivenature+ it can run in real time using only t#e present inpu t measurements andt#e previously calculated state and its uncertainty matrix no additional p astinformation is reuired!

)alman Rlter algorit#m

State model predictioneuation Pred 9 A; !;a

Prediction of state model covar iance

statePredCov 9 A;stateCov;A

prcGoise5ar

Prediction of observation mode l

covariance obsCov 9

C;statePredCov;C obsGoise5ar

Determine )alman gain

kal\ain 9 statePredCov ; C ; :obsCov=:@1=*bservation vector

prediction NPred 9C;Pred

NJrror 9 N @ NPredState estimation euation

Jst 9 Pred kal\ain ;

NJrror

Jstimated state covariance

stateCov 9 :eye:.= @ kal\ain ; C= ; statePredCov


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C-DE:

2&


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%ob0ect tracking using

kalman Rlter close all

%clear %orkspace

clear

all clc> read a video

vid*b0ect 9

5ideo'eader:car@"!avi= >

determine no! frames

n$rames 9 vid*b0ect!Gumber*f$rames

%detremine lengt# and %idt#

of frame len 9

vid*b0ect!-eig#t

%id 9 vid*b0ect!Oidt#

sum$rame 9 Neros:len+

%id=

%kalman Rlter variables

dt 9 7!1> time bet%een "

frames > initialiNe state

model matrix

A 9 B1 7 dt 7 7 1 7 dt7 7 1 77 7 7 1

9 Bdt"


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:dt&


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% iteration only

stateCov 9

prcGoise5ar

statePredCov 9 Neros:.+.=> state prediction

covariance obsGoise 9 B7!71 7!71 > noisein observation model

%variance of observation model noise+noise is gaussian innature and #ave

%Nero mean

obsGoise5ar 9 BobsGoise:1=" 77 obsGoise:"="

obsCov 9 B7 7 7 7 > initialiNe observation modelcovariance

%state vector consist of position and

velocity of ob0ect 9 B7 7 7 7

Pred 9 B7 7 7 7 > predicted

state vector Jst 9 B7 7 7 7 >

estimated state vector

N 9 B7 7 > observation vector i!e sensor

data:position of ob0ect= NPred 9 B7 7 > predicted

observation vector

%cnd is Nero till ob0ect is not detected for t#e Rrst time!

cnd 9 7

bboxPred 9

Neros:1+.=

bboxJst 9

Neros:1+.= box 9

Neros:&+.= absent

9 7

for k 9 1 Q n$rames

% read frames from

video frame 9

read:vid*b0ect+k=

gray$rame 9

rgb"gray:frame=

sum$rame 9 sum$rame double:gray$rame=


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% background obtained by averaging

sum of frames background 9

:sum$rame !< k=

ob0ect 9 false:len+%id=

% binary image consist detected ob0ect obtained bybackground subtraction ob0ect:abs:double:gray$rame= @

background=817= 9 1

% morp#ological operation to remove clutter from binaryimage

ob0ect 9 imopen:ob0ect+strel:suare+&==

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ob0ect 9

imclose:ob0ect+strel:suare+1"==

ob0ect 9 imRll:ob0ect+#oles=

% determine detected ob0ect siNe

and location blobAnalyser 9vision!lobAnalysis

blobAnalyser!MinimumlobArea 9

177

Barea+centroid+bbox 9

step:blobAnalyser+ ob0ect= if:centroid

9 7=

cnd 9 cnd

1 end% kalman algorit#m

if:cnd 99 1=

% for t#e Rrst time ob0ect detection+ initialiNestate vector

% %it# sensor observation data

:1= 9 centroid:1=

:"= 9

centroid:"=

elseif:cnd8 1=

if:centroid 9 7=

if:absent991=

:1Q"=9centroid:

1Q"= end

end

% state model prediction

euation Pred 9 A; !;a% prediction of state model

covariance statePredCov 9

A;stateCov;A prcGoise5ar

% prediction of observation model

covariance obsCov 9 C;statePredCov;C

obsGoise5ar

% determine kalman gain


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kal\ain 9 statePredCov ; C ;

:obsCov=:@1= > observation

vector prediction

NPred 9

C;Pred

if:centroid

9 7=

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N:1Q"= 9

centroid:1Q"=

NJrror 9 N @

NPred

absent 9 7 > %#en ob0ect ispresent in frame else

absent 9 1 > %#en ob0ect is

absent in frame end

% state estimation

euation Jst 9 Pred

kal\ain ; NJrror

% estimated state covariance

stateCov 9 :eye:.= @ kal\ain ; C= ;statePredCov 9 Jst

if:bbox 9 7=

bboxPred:1Q"= 9 int&":Pred:1Q"==@

:bbox:&Q.==!


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frame 9 insertS#ape:frame+ rectangle+ box+ Color+

Eblue+red+greenF= ims#o%:frame=

title:outputQblue+red+green bounded box for

observation+prediction+estimated state= subplot:&+1+"=

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ims#o%:uint4:background==

title:background=

subplot:&+1+&=

ims#o%:ob0ect=

title:binary image of detectedob0ects=

pause:7!771=

end

close all

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Experiment 1=

A%: !o st"d# -b3ect !rac*ing "sing -ptica o,.

!0E->:

(n optical o%+ motion is estimated as instantaneous image velocities or

discrete image displacements! T#e met#od calculates t#e motion bet%een t%o

video seuences t#at are taken at t%o time instances t and tt at every pixel

position! (t is based on local taylor series approxi mation of image signal i!e! t#ey

used partial derivatives %it# respect to spatial and temporal c#anges!

$or a "D@time dimension video seuence+ a pixel at location :x+ y+ t=

#aving intensity (:x+ y+ t= #ave moved x+ y and t bet%een t#e t%o video

seuenc e! (n t#is %e consider a constrain t#at t#e brig#tness remains

unaltered at di/erent position i!e!

(:x+ y+ t= 9 (:xx + yy + tt=

Assuming t#e movement to be small and using taylor series %e obtain

(:xx + yy + tt= 9 (:x+ y+ t= x y N -ig#er orderterms

T#us from above euation

x y N 9 7

9 7 T#is result in+ 9 7

O#ere and are t#e x and y components of velocity or optical o% of(:x+ y+ t= and +

, are derivatives of video seuence at :x+y+t= in t#e corresponding direction!

Oe can also use derivation of intensity ( and ( as follo% x 9y `

T#e above euation is solved to determine t#e optical o% vectors in t%odirections x and y!

$ig! 1 s#o% t#e simulink model to track a moving ob0ect in a given video! T#e

video used in t#is model is viptrac!avi consisting of moving cars on a roadscenario!

1! T#e initial block is t#e scource block+ %#ic# is used to read a multimedia Rle!

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"! T#e second block is used to convert t#e '\ values to \rayscaleintensity values!

&! T#e *ptical $lo% block is used to obtain t#e velocity vectors in x and y

direction in complex con0ugate form!

.! T#e fort# block is used t#res#olding to obtain t#e segmented ob0ects in

t#e video and to obtain t#e bounded box!

! T#e display box is used to display t#e original video+ optical o% vectors+t#e segmented ob0ects and t#e bounding box on t#e original video!

As in $ig "! t#e T#res#olding and 'egion $iltering Sub@lock initially obatins t#emagnitude of t#e velocity vector and t#en a t#res#old is computed byaveraging all t#e velocity vectors of t#e frame and using t#is t#res#old image isobtained! T#e image is t#en passed to t#e median Rlter to remove t#e noise int#e image! T#e lob Analysis Toolbox is used on t#is t#res#old image to obtaint#e bounding box of t#e ob0ect as given in $ig &!

As in $ig .!t#e display box consist of . inputsQ input video+ t#res#olded video+optical o% vectors and t#e tracked output result!

T#e input video and t#e t#res#old image is directly displayed! T#e optical o%vectors are passed t#roug# optical o% line function block to give t#edimensions of t#e optical o% vectors! T#ese dimensions are t#en ploted on t#eoriginal video and displayed! T#e bounding box are counted to obtain t#enumber of cars and t#e bounding box is diplayed on t#e original video to obatint#e result!

S%


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$ig!" T#res#olding and 'egion $ilteringSubblock

$ig!& 'egion $ilteringSubblock


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$ig!. Display 'esult Subblock

-

3"


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(a) (b)

(c) (d)

(e)

$ig! :a= Gt# $rame of t#e video :b=G1t# $rame of t#e video :c= *ptical o%velocity vectors for frame :a= and :b= + :d= T#res#olding for ob0ect detection and:e= *b0ect detection


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3&

anadi anant jain

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