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CPSC 452: Lecture 1 Introduction, Homogeneous transformations and Coordinate frames

CPSC 452: Lecture 1 Introduction, Homogeneous transformations and Coordinate frames

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CPSC 452: Lecture 1

Introduction,

Homogeneous transformations and Coordinate frames

2

Introduction

• Robots in movie

3

Modern Robots

• Robot in life– Industry– Medicare

4

Modern Robots

• Robot in life– Home/Entertainment

5

Modern Robots

• Robots in life– Military/Unmanned Vehicle

6

What is a robot

• “A robot is a reprogrammable multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks”– by Robot Institute of America

7

Scope of CPSC 452

Planning Sensing

Control

Dynamics

Kinematics

Rigid body mechanics

8

Scope of CPSC 452

Planning Sensing

Control

Dynamics

Kinematics

Rigid body mechanics

9

Intro

• Space– Type – Physical, Geometry, Functional– Dimension– Direction

• Basis vectors

– Distance• Norm

– Description – Coordinate System• Matrix

10

A review of vectors and matrix

• Vectors– Column vector and row vector

– Norm of a vector

nv

v

v

v2

1

nvvvv 21

222

21 ... nvvvv

11

Dot product of two vectors

• Vector v and w

• If |v|=|w|=1,

cos|||| wvwv

cos wv

v

w

12

Position Description

• Coordinate System A

z

y

x

PA

PA

AX

AY

AZ

13

Orientation Description

• Coordinate System A

z

y

x

PA

PA

AX

AY

AZ

14

Orientation Description

• Coordinate System A• Attach Frame

Coordinate System B

z

y

x

PA

PA

AX

AY

AZBZ

BYBX

15

Orientation Description

• Coordinate System A• Attach Frame

Coordinate System B• Rotation matrix

PA

AX

AY

AZBZ

BYBX

333231

232221

131211

ˆˆˆ

rrr

rrr

rrr

ZYXR BA

BA

BAA

B

AB XXr ˆˆ11

AB YXr ˆˆ21

AB ZXr ˆˆ31

16

Rotation matrix

ABABAB

ABABAB

ABABAB

BA

BA

BAA

B

ZZZYZX

YZYYYX

XZXYXX

ZYXR

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆ

BA X

TA

B X

TBA

T

AB

AB

AB

TA

B

TA

B

TA

B

RZYX

Z

Y

X

ˆˆˆ

ˆ

ˆ

ˆ

Directional Cosines

Directional Cosines

17

Rotation matrix

• For matrix M,– If M-1 = MT , M is orthogonal matrix– is orthogonal!!

1 RRIRR BA

AB

BA

AB

TBA

AB RR

1 RR BA

TAB

RAB

18

Orthogonal Matrix

ABABAB

ABABAB

ABABAB

BA

BA

BAA

B

ZZZYZX

YZYYYX

XZXYXX

ZYXR

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆ

9 Parameters to describe orientation!

19

Description of a frame

• Position + orientation

BORGAP

AX

AZBZ

BYBX},{}{ BORG

AAB PRB

AY

20

Graphical representation

BORGAP

AX

AZBZ

BYBX

AY{A}

{B}

{U}

uZ

uX

uY

},{}{ AORGuu

A PRA

},{},{}{ BORGAA

BBORGuu

B PRPRB

BORGuP

21

Mapping – Translation Difference

BORGAP

AX

AZBZ

BYBX

AY

PB

PPP BBORG

AA

PA

11

PT

P BAB

A

1000

BORGA

AB

PIT

22

Mapping – rotation difference

AX

AZ

AY

BZ

BY

BX

PB

PR

P

P

P

ZYX

ZPYPXPP

ZPYPXP

P

P

P

P

BAB

z

y

x

B

BA

BA

BA

BAzB

AyB

Ax

A

BzByBx

z

y

x

B

B

]ˆˆˆ[

ˆˆˆ

ˆˆˆPRP BA

BA

11

PT

P BAB

A

1000

0RT

ABA

B

23

ExampleBY

BX

AXAYPB

30

30

AB ZZ ˆˆ

?

1

2

1

PP AB

0cos90cos90cos

90cos30cos120cos

90cos60cos30cos

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆ

ABABAB

ABABAB

ABABAB

BA

BA

BAA

B

ZZZYZX

YZYYYX

XZXYXX

ZYXR

24

PBPRBAB

Rotation + Translation Difference

BORGAP

AZBZ

BY

BX

AY

PA

AXPA

25

Homogeneous Transformation

PRPP BABBORG

AA

110001_

_

_

zB

yB

xB

zBORGA

yBORGAA

B

xBORGA

zAy

Ax

A

P

P

P

P

PR

P

P

P

P

TAB

26

Mapping – Translation Difference

BORGAP

AX

AZBZ

BYBX

AY

PB

PPP BBORG

AA

PA

11

PT

P BAB

A

1000

BORGA

AB

PIT

27

Translation Operator

1PA

AX

AZ

AY

QPP AAA 12QA

• Translation operator2P

A

12 )( PqDP AQ

A

1000

100

010

001

)(z

y

x

Q q

q

q

qD

222zyx qqqq

28

Mapping – rotation difference

AX

AZ

AY

BZ

BY

BX

PB

PRP AAB

A

11

PT

P BAB

A

1000

0RT

ABA

B

29

Rotation Operator

AX

AZ

AY

1PA

2PA

BX

BY

21 PP AB

100

0cossin

0sincos

0cos90cos90cos

90coscos)90cos(

90cos)90cos(cos

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆ

ABABAB

ABABAB

ABABAB

BA

BA

BAA

B

ZZZYZX

YZYYYX

XZXYXX

ZYXR