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

  • 5

    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