review: algebra and geometry
TRANSCRIPT
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
