kinematics 1: fundamentals of orthogonal robot systems alonzo kelly january 12, 2005 1 kinematics...

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  • Alon

    1

    K

    tics is Importantg is one of the mostools of robotics.

    g:

    ontroline programming and

    s specifically, you needr:

    trol, simulation)age formation)

    d and communicat ionging (maps, object models)ing (motion prediction)f legged vehicles

    zo Kelly January 12, 2005

    perception (im sensor heaantenna pointin world modell terrain follow gait control o

    Mobile Robot Systems

    inematics 1: Fundamentals of Orthogonal Transforms

    1 Why KinemaKinematic modellinimportant analytical t

    Used for modellin mechanisms actuators sensors

    Used for on-line c Used for off - lsimulation In mobile robotkinematic models fo

    steering (con

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms2

    2 2.1

    n

    tn

    2.2

    to represent:hich change the entity)s (wh ich cha nge t he

    Frames (which represent

    perscripts

    pts such as the 1 in ity from another, such as

    ripts such as the in ate system within whichere) is represented. Theseoordinate system frome same point.

    rm of shorthand for trig

    p1

    b pb

    1 2 3+ +( )1)

    zo Kelly January 12, 2005

    Points, Operators, Etc.

    Vectors are used to represent points.

    T tyx tyy tyztzx tzy tzz

    = 2.4 Trig Functions

    We often use a fofunctions:

    c123 cos=

    s1 (sin=

    Mobile Robot Systems 2 Notational Conventions2.1Vectors and Matrices

    Notational Conventions Vectors and Matrices

    Vectors are represented as columns ofumbers:

    Sometimes will be written to emphasizehat it is a vector. Matrices are represented by an array ofumbers:

    p x y zT=

    p p

    txx txy txz

    Matrices are used Operators (w Tr ans fo rmrepresentation) Coordinate rigid bodies)

    2.3 Subscripts and su

    Leading subscridistinguish one ent

    . Leading superscdenotes the coordinthe object (a point hdistinguish one canother - even for th

    p2

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms3

    2.5

    vv

    2.6

    mee

    t ion i s such tha t thebscripts provide a hint ofing that the as cancel:

    Tabpa=

    zo Kelly January 12, 2005

    Converting Coordinates

    Suppose the 4 X 4 matrix denotes theatrix which transforms a vector from its

    xpression in coordinate system a to itsxpression in coordinate system b.

    xa Jacobian tensor

    Tab

    Mobile Robot Systems 2 Notational Conventions2.5Derivatives

    Derivatives

    All combinations of derivatives of (scalars,ectors, matrices) with respect to (scalars,ectors, matrices) are defined and meaningful.

    xy

    xy

    x

    y

    a partial derivative

    a gradient vector

    a vector partial derivative

    x

    y

    Y

    a Jacobian matrix

    xY a matrix partial derivative

    Then , t he no tasuperscripts and suthe result by imagin

    pb

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms4

    3 igi

    w

    formation preserves linesry transformation). That points are on a line, their on a line.

    preserve the distance, and hence, it may notr angles between lines.2 transformation is ann for which .

    ormation does not includedoes include all the others

    ansformation transformsctangular coordinates to

    are mutually orthogonalgth, thus.

    is linear, the equation b Ax 0=

    t1 t2 0= =

    r11 r12r21 r22

    x1y1

    zo Kelly January 12, 2005

    scale reflections shear.

    1. see Encyclopedia of Mathematics, ed. by I. M. Vinogradov, Kluwer, 1988 and/or Mathematics Dictionary, Van Nostrand, 1992.

    an orthogonal trfrom one set of reanother.

    Its columns and of unit len

    2. While the equation is homogeneous.

    Ax =

    Mobile Robot Systems 3 Definitions2.6Converting Coordinates

    Definitions consider linear relationships between pointsn 2D. an affine transformation1 is the mosteneral linear transformation possible. In 2Dt looks like:

    here the rs and ts are constants. Such a transformation includes:

    translation rotation

    x2y2

    r11 r12r21 r22

    x1y1

    t1t2

    +=

    Such a trans(aka collineatois, if two inputoutputs are also It may notbetween thempreserve area o

    a homogeneousaffine transformatio

    Such a transftranslation but above.

    x2y2

    =

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms5

    tion

    neous Transformsa matrix and a vector is matrix is a mapping fromectors. of a matrix is that of an

    s which maps points onto

    projective coordinates are property that the object does not change if all of

    e multiplied by the sameThey are unique up to a

    ordinates are a method oftities by the projections of3D subspace. Followingy you would use such an

    h Translation

    ply a 3D point by the most

    zo Kelly January 12, 2005

    points establish whawkward construct.

    4.1 The Problem wit

    Suppose we multigeneral 3X3 matrix:

    Mobile Robot Systems 4 Why Homogeneous Transforms4.1The Problem with Transla

    Such a transformation preserves thedistance between two points, so it isrigid. It therefore also preserves lengths,areas, and angles. It does not scale, reflect, or shear. Onlyrotation is left.

    r11r12 r21r22+ 0=

    r11r11 r21r21+ 1=

    r12r12 r22r22+ 1=

    4 Why Homoge The product of another vector. So avectors onto other v So......... One viewoperator - a procesother points. Homogeneous or those having thedetermined by themthe coordinates arnonzero number. scale factor. Homogeneous corepresenting 3D en4D entities onto a

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms6

    Coordinates

    opetsv

    epresented as a l ineare elements of - it ispendent of .ogeneous Coordinates

    be fixed with a standardeous coordinates, (akates) an extra element, ,

    h point to represent a kind

    nal to consider that this 4- into 3D by dividing by

    y represented with a scale

    p1p1

    w1

    1 y1 z1 w1T

    x1w1------

    y1w1------

    z1w1------

    T

    1 y1 z1 1T

    zo Kelly January 12, 2005

    imple addition of a constant vector to anotherector like:

    p2 p1 pk+

    x1y1z1

    xkykzk

    += = Points are typicallfactor of 1. Thus:

    p1 =

    p1 x=

    Mobile Robot Systems 4 Why Homogeneous Transforms4.2Introducing Homogeneous

    This most general transform can representperators like scale, reflection, rotation, shear,rojection. Why? because all of these can bexpressed as constant linear combinations ofhe coordinates of the input vector. However, this 3X3 matrix cannot represent a

    p2

    x2

    y2

    z2

    Tp1= =

    Tp1

    txx txy txztyx tyy tyz

    tzx tzy tzz

    x1y1

    z1

    txxx1 txyy1 txzz1+ +

    tyxx1 tyyy1 tyzz1+ +

    tzxx1 tzyy1 tzzz1+ +

    = =

    cannot be rcombination of thsupposed to be inde

    4.2 Introducing Hom

    The situation cantrick. In homogenprojective coordinacan be added to eacof scale factor:

    and it is conventiovector is projectedthe scale factor:

    p2

    p1 x=

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms7

    eous Transforms

    iNcdu

    ic

    4.3

    ipv

    i s t h e homogeneousequivalent to a translationslation vector . sforms can also representear operations such ason. They do this by hiding the normalization by the

    x1

    y1

    z1

    1

    xk

    yk

    zk

    1

    +

    Trans pk( )p1

    = =

    =

    pk

    zo Kelly January 12, 2005

    Using homogeneous coordinates, it is nowossible to represent the addition of twoectors as a matrix operation, thus:

    the nonlinearity inscale factor.

    Mobile Robot Systems 4 Why Homogeneous Transforms4.3Translation with Homogen

    s a point in homogeneous coordinates.otationally, is the homogeneous point

    orresponding to . It is also possible to represent a pureirection in terms of a point at infinity bysing a scale factor of 0. Thus:

    s a direction in homogeneous coordinates. From now on, we will drop the ~ and useontext to distinguish these. Translation with Homogeneous

    Transforms

    A (3D) matrix in homogeneous coordinatess a 4X4 matrix.

    p1p1

    q1 x1 y1 z1 0T

    =

    where transform which is operator for the tran Homogeneous transomewhat nonlinperspective projecti

    p2 p1 pk+

    1 0 0 xk0 1 0 yk0 0 1 zk0 0 0 1

    x1

    y1

    z1

    1

    =

    Trans pk( )

  • Alon

    Kinematics 1: Fundamentals of Orthogonal Transforms8

    as Operators

    5 Thho- tint

    5.1

    sesoHcsttop

    that they are orthogonal3 rows and columns arel.ntified (here) by an upper

    Trans u v w, ,( )

    1 0 0 u

    0 1 0 v

    0 0 1 w

    0 0 0 1

    =

    Rotx ( )

    1 0 0 0

    0 c s 00 s c 00 0 0 1

    =

    al Orthogonal Operators -ous vector is multiplied by

    the result is a new vector thatnslated.

    zo Kelly January 12, 2005

    The basic orthogonal operators areranslation along and rotation about any of thehree axes. The following four elementaryperators are sufficient for almost any realroblems.

    y

    Figure 4 FundamentPart 1. If a homogeneone of these matrices, has been rotated or tra

    Mobile Robot Systems 5 Semantics and Interpretations5.1Homogeneous Transforms

    Semantics and Interpretationse whole trick of successful manipulation ofmogeneous coordinates is to get the mindsethe semantics associated with their manyerpretations.

    Homogeneous Transforms as Operators

    Suppose it is necessary to move a point inome manner to generate a new point andxpress the result in the same

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