review: algebra and geometry

Lecture 4

Mobot Kinematics and Control

CSE390/MEAM420-520

Some notes taken from Siegwart&Nourbakhsh

Review: Algebra and Geometry

Vectors• Ordered set of numbers: (1,2,3,4)

• Example: (x,y,z) coordinates of pt in space.

runit vecto a is ,1 If

),(

1

2

,,21

vv

xv

xxxv

n

i i

n

=

=

=

!=

K

Distance = norm

x1

x2

Vector Addition

),(),(),( 22112121 yxyxyyxx ++=+=+wv

vvww

V+wV+w

Scalar Product

),(),( 2121 axaxxxaa ==v

vv

avav

Inner (dot) Product

vv

ww

!!

22112121 .),).(,(. yxyxyyxxwv +==

The inner product is a The inner product is a SCALAR!SCALAR!

Angle:

Unit circle, angle

Inner (dot) Product

vv

ww

!!

22112121 .),).(,(. yxyxyyxxwv +==

!cos||||||||),).(,(. 2121 wvyyxxwv "==

wvwv !"= 0.

Angle:

< = > V,W are independent of each other

Matrices

!!!!!!

"

#

$$$$$$

%

&

='

nmnn

m

m

m

mn

aaa

aaa

aaa

aaa

A

L

MOMM

L

L

L

21

33231

22221

11211

mnmnmnBAC !!! +=

Sum:Sum:

ijijij bac +=

A and B must have the sameA and B must have the same

dimensionsdimensions

Matrices

pmmnpn BAC !!! =

Product:Product:

!=

=m

k

kjikij bac1

A and B must haveA and B must have

compatible dimensionscompatible dimensions

nnnnnnnnABBA !!!! "

Is matrix multiplication associative?

Matrices

pmmnpn BAC !!! =

Product:Product:

!=

=m

k

kjikij bac1

A and B must haveA and B must have

compatible dimensionscompatible dimensions

nnnnnnnnABBA !!!! "

Identity Matrix:AAIIAI ==

!!!!!

"

#

$$$$$

%

&

=

100

010

001

O

OOOO

O

O

Matrices

mnT

nmAC !! =

Transpose:Transpose:

jiijac = TTT

ABAB =)(

TTTBABA +=+ )(

IfIf AAT= A is symmetricA is symmetric

Dot product:

Scaling Equation

PP

xx

yy

s.xs.x

PP’’s.ys.y

),('

),(

sysx

yx

=

=

P

P

PP != s'

!"

#$%

&'!"

#$%

&=!

"

#$%

&(

y

x

s

s

sy

sx

0

0'P

SPSP !='

Rotation

PP

PP’’

Rotation Equations

Counter-clockwise rotation by an angle Counter-clockwise rotation by an angle ""

!"

#$%

&!"

#$%

& '=!

"

#$%

&

y

x

y

x

((

((

cossin

sincos

'

'

PP

xx

YY’’PP’’

""

XX’’

yy R.PP'=

Degrees of Freedom

R is 2x2 R is 2x2 4 elements4 elements

BUT! There is only 1 degree of freedom: BUT! There is only 1 degree of freedom: ""

1)det( =

=!=!

R

IRRRRTT

The 4 elements must satisfy the following constraints:The 4 elements must satisfy the following constraints:

!"

#$%

&!"

#$%

& '=!

"

#$%

&

y

x

y

x

((

((

cossin

sincos

'

'

Stretching Equation

PP

xx

yy

SSxx.x.x

PP’’SSyy.y.y

!"

#$%

&'!"

#$%

&=!

"

#$%

&(

y

x

s

s

ys

xs

y

x

y

x

0

0

'P

),('

),(

ysxs

yx

yx=

=

P

P

S

PSP !='

Matrices

Determinant:Determinant: A must be squareA must be square

3231

2221

13

3331

2321

12

3332

2322

11

333231

232221

131211

detaa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

+!=

"""

#

$

%%%

&

'

12212211

2221

1211

2221

1211det aaaa

aa

aa

aa

aa!=="

#

$%&

'

Volume = determinant

Cross product

X is normal to

both v and w

Matrices

IAAAAnn

nnnnnn

== !!"

!"

!

11

Inverse:Inverse: A must be squareA must be square

!"

#$%

&

'

'

'=!

"

#$%

&'

1121

1222

12212211

1

2221

1211 1

aa

aa

aaaaaa

aa

Kinematics Models of wheel (rolling and sliding contraints)

Mobil robot(mobot) maneuverability(possible space and velocity it can reach)

Mobot Kinematic Control(Sequence of movements toward target)

Rotation:

Transformation to the

Reference frame

Motion caused by wheel motion, rolling constraints:

Motion in the orthogonal plane must be 0, sliding constraints:

Kinematics Models of wheel (rolling and sliding contraints)

Mobil robot(mobot) maneuverability(possible space and velocity it can reach)

Mobot Kinematic Control(Sequence of movements toward target)

Examples: differential drive + omidirectional drive

ICR is the Null space of the C1

ICR is the Null space of the C1; Null(C1) + Rank(C1) = 3

Palm Pilot Robot, CMU

Kinematics Models of wheel (rolling and sliding contraints)

Mobil robot(mobot) maneuverability(possible space and velocity it can reach)

Mobot Kinematic Control(Sequence of movements toward target)

Why there is a “S” curve shape to the path?

Kinematics Models of wheel (rolling and sliding contraints)

Mobil robot(mobot) maneuverability(possible space and velocity it can reach)

Mobot Kinematic Control(Sequence of movements toward target)

Optional Slides, will return to

this in path planning lectures.

Degree of Freedom DOF= possible space (x,y,theta) a robot can reach

Is Degree of Freedom of robot same as its maneuverability?

What is the DOF of the Ackerman vehicle?

Differentiable Degree of Freedom DDOF= possible velocity a robot can reach

DDOF = degree of Mobility,

Omnibot:

DDOF = 3; DOF = 3.

Robot is Robot is HolonomicHolonomic <=> DDOF = DOF <=> DDOF = DOF