# Lecture 1 Rigid Body Transformations

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• Lecture 1Rigid Body Transformations

Joaquim SalviUniversitat de Girona

Visual Perception

http://www.vibot.org/http://www.vibot.org/http://europa.eu.int/comm/education/programmes/mundus/index_en.htmlhttp://europa.eu.int/comm/education/programmes/mundus/index_en.html

• 2

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 3

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 4

Lecture 1: Rigid Body Transformations

Standard base vectors:

1.1 Cartesian coordinates, Points and Vectors

Coordinate System: Complete set of orthonormal vectors (perpendicular and unit) and coinciding in a point (origin).

Reference System: Unique/World coordinate system used for referencing points, vectors and other coordinate systems

right-hand frame

{A}

Ax

Ay

Az

1

0

0

AX

0

1

0

AY

0

0

1

AZ

O

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Lecture 1: Rigid Body Transformations

Coordinates of a point in space:

zpzpc

ypypb

xpxpa

zcybxa

c

b

a

p

T

T

T

{A}

Ax

Ay

Az

a

b

c

pA u

Ap

Referenced with respect to {A}

Corresponding to object/reference u

1.1 Cartesian coordinates, Points and Vectors

• 6

Lecture 1: Rigid Body Transformations

A free vector is defined by a pair of points :

Coordinates of the vector : {A}

Ax

Ay

Az

O

pA q

A v

A

vA

1

2

3

Ap

p p

p

1

2

3

Aq

q q

q

1 1 1

2 2 2

3 3 3

Av q p

v v q p

v q p

1.1 Cartesian coordinates, Points and Vectors

• 7

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 8

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 9

Lecture 1: Rigid Body Transformations

Inner product between two vectors:

1.2 Inner product and Cross product

• 10

Lecture 1: Rigid Body Transformations

Antisymmetric matrix

1.2 Inner product and Cross product

Cross product between two vectors:

• 11

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 12

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 13

Lecture 1: Rigid Body Transformations

{B} is a translation of {A} to position A(3 4 3)T.

Coodinate axes of {B} are paralel to coordinate axes of {A}.

The translation can be representated by vectorial addition.

5

7

4

3

4

3

2

3

1

;

AAB

z

A

y

A

x

A

z

A

y

A

x

A

z

B

y

B

x

B

z

A

y

A

x

A

z

B

y

B

x

B

z

A

y

A

x

A

p

p

p

o

o

o

p

p

p

o

o

o

p

p

p

p

p

p

Bp=(1 3 2) T

Ap=(4 7 5) T

AoB=(3 4 3)T

{A}

Az Bz

Ay

By

AxBx

{B}

Translation vector:

1.3 Translations

• 14

Lecture 1: Rigid Body Transformations

Coordinates are related by:

p )oT( =pB

B

AA

B

A

B

z

A

y

A

x

AA

B

AtI

o

o

o

oT

10

1000

100

010

001

)(

B

ABAo p= p

Translation matrix:

1.3 Translations

• 15

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 16

Lecture 1: Rigid Body Transformations

Contents

1. Rigid Body Transformations1.1 Cartesian coordinates, Points and Vectors.

1.2 Inner product and Cross product

1.3 Translations

1.4 Rotations

1.5 Homogeneous coordinates

1.6 Composition of transformations

1.7 Inverse transformation

1.8 Parametrization of a Rotation matrix

1.9 Extracting the rotation angles of a Rotation matrix

1.10 Computing the closest rotation matrix of a noisy rotation matrix

• 17

Lecture 1: Rigid Body Transformations

Let {A} and {B} two ortonormal coordinate systems with

the same origin and unit vectors {a1, a2, a3} {b1, b2, b3}.

b1

a1

a2

a3

b2b3

332211

332211

321

B

321

A

{B} and {A}in by vector drepresente is Point

bpbpbpp

apapapp

pppp

pppp

pp

ABABABA

AAAAAAA

TBBB

TAAA

3332321313

3232221212

3132121111

3

1

3

1

3

1

3

2

1

)(

Consider

prprprpk

prprprpk

prprprpk

rpabpabpapp

p

BBBA

BBBA

BBBA

kj

j

j

B

j

k

A

j

A

j

B

k

A

j

A

j

B

j

k

AA

k

A

k

A

3

2

1

332313

322212

312111

3

2

1

p

p

p

bababa

bababa

bababa

p

p

p

B

B

B

B

A

A

A

A

Rotation matrix:

1.4 Rotations

• 18

Lecture 1: Rigid Body Transformations

Rotation matrix:

332313

322212

312111

bababa

bababa

bababa

RBA

b1

a1

a2

a3

b2b3

1bA

2bA

3bA

332313

322212

312111

bababa

bababa

bababa

RBA

1aB

2aB

3aB

ABT

B

ARR

, det( ) 1, rank( ) 3T

R R I R R

1.4 Rotations

• 19

Lecture 1: Rigid Body Transformations

Basic Rotation matrices:

cossin0

sincos0

001

332313

322212

312111

B

A

bababa

bababa

bababa

cos0sin

010

sin0cos

332313

322212

312111

B

A

bababa

bababa

bababa

Ro

t(, x

)R

ot(, y)

Y

X

ZZ

X

Y

X

Z

YZ

Ro

t(, z

)

Y

X

ZZ

X

Y

100

0cossin

0sincos

332313

322212

312111

B

A

babab

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