Basic Kinematics

Basic Kinematics

•Spatial Descriptions and Transformations•Introduction to Motion

Objectives of the Lecture

• Learn to represent position and orientation• Be able to transform between coordinate

systems.• Use frames and homogeneous coordinates

Reference: Craig, “Introduction to Robotics,” Chapter 2.Reference: Craig, “Introduction to Robotics,” Chapter 2.

Handout: Chapter 1Handout: Chapter 1

Almost any introductory book to roboticsAlmost any introductory book to robotics

Introduction

• Robot manipulation implies movement in space

• Coordinate systems are required for describing position/movement

• Objective: describe rigid body motion• Starting point: there is a universe/ inertial/

stationary coordinate system, to which any other coordinate system can be referred

The Descartes Connection • Descartes invented what we now call Cartesian

coordinates, or the system by which we can graph mathematical functions in two-or three-dimensional space.

• So, all of those problems you have been working in algebra are his fault.

• Descartes was lying on his bed watching a fly. Slowly, it came to him that he could describe the fly's position at any instant by just three numbers. Those three numbers were along the planes of the floor and two adjacent walls, what we now call the x,y,z coordinate system.

• So, all of those problems you have been working in algebra are really the fly’s fault (kill that fly)

Coordinate System in Robotics

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Three Problems with CS

• Given 2 CS’s, how do we express one as a function of the other?

• Given a point in one CS, what are the point’s coordinates on a second CS?

• Apply an operation on a vector

Description of a Position

• point = position vector

YA

XA

ZA

{A}

AP

z

y

xA

ppp

P

a coordinate system

Description of an Orientation

• In the example, a description of {B} with respect to {A} suffices to give orientation

• Orientation = System of Coordinates

• Directions of {B}: XB, YB and ZB

• In {A} coord. system: AXB, AYB and AZB

x0

x1

z0

y0

z1

y1

{A}

{B}

Often a point is not enough: need orientation

From {A} to {B}{A}

XB

aXaY

aZ BAX XX cos BAY XY cos

BAZ XZ cos

We conclude:

BA

BA

BA

BA

XZXYXX

X

Rotation Matrix• Stack three unit vectors to form Rotation Matrix• describes {B} with respect to {A}

• Each vector in can be written as dot product of pair of unit vectors: cosine matrixcosine matrix

• Rows of : unit vectors of {A} with respect to {B}

• What is ? What is det( )?• Position + orientation = Frame

RAB

BA

BA

BAA

B ZYXR ˆˆˆRA

B

RAB

1RAB RA

B

Description of a Frame• Frame: set of four

vectors giving position + orientation

• Description of a frame: position + rotation matrix

• Ex.:

x0

x1

z0

y0

z1

y1

{A}

{B}

BORGAA

B PRB ,}{

• position: frame with identity as rotation

• orientation: frame with zero position

Mapping: from frame 2 frame

• If {A} has same orientation as {B}, then {B} differs from {A} in a translation: APBORG

AP = BP + APBORG

• Mapping: change of description from one frame to another. The vector APBORG defines the mapping.

XA

ZA

YA

{A}ZB

YB

{B}

XB

AP

APBORG

BP

Translated Frames

Rotated Frames

YA

XA

ZA

X B

Z B

YB

BPBzByBx

B ZpYpXpP

BA

zBA

yBA

xA ZpYpXpP

z

y

x

BA

BA

BAA

ppp

ZYXP

PRP BAB

A

Description of Rotation = Rotation Matrix

Rotated Frame (cont.)• The previous expression can be written as

• The rotation mapping changes the description of a point from one coordinate system to another

• The point does not change! only its description

PRP BAB

A

Example (2D rotation)

XA

YA

Y B

X B

x0

y0

x 1

y 1

cossinsincos

001

001

yxyyxx

General Frame Mapping

{A}

XB

YB

ZB

BP

XA

YA

ZA

AP

APBORG

BORGABA

BA PPRP

Replace by the more appealing equation:

110001PPRP B

BORGAA

BA

A “1” added here A row added here

Homogeneous Coords• Homogeneous coordinates: embed 3D

vectors into 4D by adding a “1”• More generally, the transformation matrix T

has the form:

FactorScalingTrans. Perspect.Vector Trans. MatrixRot.

T

Operators: Translation, Rotation and General Transformation

• Translation Operator:

1

12

|)|,ˆ( PQQTRANS

QPPAA

AAA

1P

2PQ

Translation Operator

• Translation Operator: • Only one coordinate frame, point moves• Equivalent to mapping point to a 2nd frame• Point Forward = Frame Backwards

• How does TRANS look in homogeneous coordinates?

• Rotational OperatorOperators (cont.)

AP1

AP2

K

Rotation around axis:

Rotation Operator

• Rotational OperatorThe rotation matrix can be seen as rotational

operator• Takes AP1 and rotates it to AP2=R AP1

• AP2=ROT(K, q)(AP2) • Write ROT for a rotation around K

Operators (Cont.)• Transformation Operators* A transformation mapping can be viewed as

a transformation operator: map a point to any other in the same frame

* Transform that rotates by R and translates by Q is the same a transforming the frame by R & Q

Compound TransformationIf {C} is known relative to {B}, and {B} is

known relative to {A}. We want to transform P from {C} to {A}:

Write down the compound in homog. coords

TTT

PTP

PTP

PTP

BC

AB

AC

BAB

A

CAC

A

CBC

B

define here From

Inverse Transform

Write down the inverse transform in HC’s

More on Rotations• We saw that a rotation can be represented by

a rotation matrix• Matrix has 9 variables and 6+ constraints

(which?)• Rotations are far from intuitive: they do not

commute!• Rotation matrix can be parameterized in

different manners:—Roll, pitch and yaw angles—Euler Angles—Others

Euler’s Theorem

• Euler’s Theorem: Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.

• Not to be confused with Euler angles, Euler integration, Newton-Euler dynamics, inviscid Euler equations, Euler characteristic…

• Leonard Euler (1707-1783)

XYZ Fixed

Euler Angles• This means that we can represent an orientation with 3

numbers• A sequence of rotations around principle axes is called an

Euler Angle Sequence• Assuming we limit ourselves to 3 rotations without

successive rotations about the same axis, we could use any of the following 12 sequences:

XYZ XZY XYX XZXYXZ YZX YXY YZYZXY ZYX ZXZ ZYZ

Euler Angles

• This gives us 12 redundant ways to store an orientation using Euler angles

• Different industries use different conventions for handling Euler angles (or no conventions)

ZYX Euler

Euler Angles to Matrix Conversion

• To build a matrix from a set of Euler angles, we just multiply a sequence of rotation matrices together:

yxzxzyxzxzyx

yxzxzyxzxzyx

yzyzy

cccssscsscsccsccssssccssssccc

10000

0010

0

00

001

zz

zz

yy

yy

xx

xxzyx cssc

cs

sc

csscRRR

Euler Angle Order

• As matrix multiplication is not commutative, the order of operations is important

• Rotations are assumed to be relative to fixed world axes, rather than local to the object

• One can think of them as being local to the object if the sequence order is reversed

Using Euler Angles

• To use Euler angles, one must choose which of the 12 representations they want

• There may be some practical differences between them and the best sequence may depend on what exactly you are trying to accomplish

x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Euler Angle I, Animated

Orientation Representation• Euler Angle I

1000cossin0sincos

,cossin0sincos0001

,1000cossin0sincos

''

'

w

uz

R

RR

Euler Angle I

cossincossinsin

sincoscoscoscos

sinsincossincos

cossin

sinsincoscossin

sincoscossinsin

coscos''' wuz RRRR

Resultant eulerian rotation matrix:

Euler Angle II, Animated

x

y

z

u'

v'

=v"

w"

w'=

u"

v"'

u"'

w"'=

Note the opposite (clockwise) sense of the third rotation, .

Orientation Representation• Matrix with Euler Angle II

cossinsinsincos

sinsincoscossin

coscoscoscossin

sincos

sincoscoscossin

cossincoscoscos

sinsin

Quiz: How to get this matrix ?

Orientation Representation• Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?

Vehicle Orientation

• Generally, for vehicles, it is most convenient to rotate in roll (z), pitch (x), and then yaw (y)

• In situations where thereis a definite ground plane,Euler angles can actuallybe an intuitiverepresentation x

y

z

front of vehicle

Euler Angles - Summary

• Euler angles are used in a lot of applications, but they tend to require some rather arbitrary decisions

• They also do not interpolate in a consistent way (but this isn’t always bad)

• There is no simple way to concatenate rotations• Conversion to/from a matrix requires several

trigonometry operations• They are compact (requiring only 3 numbers)

Quaternions

Quaternions

• Quaternions are an interesting mathematical concept with a deep relationship with the foundations of algebra and number theory

• Invented by W.R.Hamilton in 1843• In practice, they are most useful to us as a means

of representing orientations• A quaternion has 4 components

3210 qqqqq

Quaternions (Imaginary Space)

• Quaternions are actually an extension to complex numbers• Of the 4 components, one is a ‘real’ scalar number, and the

other 3 form a vector in imaginary ijk space!

3210 kqjqiqq q

jiijkikkijkjjki

ijkkji

1222

Quaternions (Scalar/Vector)

• Sometimes, they are written as the combination of a scalar value s and a vector value v

where

321

0

qqqqs

v

vq ,s

Unit Quaternions• For convenience, we will use only unit length quaternions,

as they will be sufficient for our purposes and make things a little easier

• These correspond to the set of vectors that form the ‘surface’ of a 4D hypersphere of radius 1

• The ‘surface’ is actually a 3D volume in 4D space, but it can sometimes be visualized as an extension to the concept of a 2D surface on a 3D sphere

123

22

21

20 qqqqq

Quaternions as Rotations

• A quaternion can represent a rotation by an angle θ around a unit axis a:

• If a is unit length, then q will be also2

sin,2

cos

2sin

2sin

2sin

2cos

aq

q

or

aaa zyx

Quaternions as Rotations

112

sin2

cos2

sin2

cos

2sin

2cos

2sin

2sin

2sin

2cos

22222

22222

2222222

23

22

21

20

a

q

zyx

zyx

aaa

aaa

qqqq

Basic Kinematics

Documents

flys position

z coordinate system

flycoordinate system

position rotation matrixex

position vectoryadescription

frame description

flys fault

unit vectors