Jinxiang Chai Composite Transformations and Forward Kinematics 0.

131
Jinxiang Chai Composite Transformations and Forward Kinematics 1

Transcript of Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Page 1: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Jinxiang Chai

Composite Transformations and Forward Kinematics

1

Page 2: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Outline

2D Coordinate transformation

Composite transformation & Forward kinematics

3D transformation

2

Page 3: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Image space

Coordinate Transform: 3D Geometry Pipeline

3

Normalized project

space

View spaceWorld spaceObject space

Aspect ratio &

resolution

Focal length

Rotate and translate the

camera

Page 4: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation: 3D Modeling/Design

Coordinate transformation from one reference frame to another

4

Page 5: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation: Animation/Robotics

How to model 2D movement of animated characters or robots?

5

Click here

Page 6: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation

Coordinate transformation from one reference frame to another

6

Page 7: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation

Coordinate transformation from one reference frame to another

7

Page 8: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation

Coordinate transformation from one reference frame to another

8

Page 9: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Coordinate Transformation

Coordinate transformation from one reference frame to another

?

9

Page 10: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product

v w

10

Page 11: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product: measuring similarity between two vectors

v w

)cos(|||| wvwv

11

Page 12: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product: measuring similarity between two vectors

v w

)cos(|||| wvvw

12

Page 13: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product: measuring similarity between two vectors

v w

1vvUnit vector:

13

Page 14: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product: measuring similarity between two vectors

v w

)cos(|||| wvwv

14

Page 15: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Dot Product: measuring similarity between two vectors

v w

0wv

15

Page 16: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Cross Product: measuring the area determined by two vectors

16

v w

Page 17: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Review – Vector Operations

Cross Product: measuring the area determined by two vectors

sin 2*v w v w u area u

17

v w

u

Page 18: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinates

2D Cartesian coordinate system:

P: (x,y)

i

j

o

18

Page 19: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D Cartesian coordinate system:

P: (x,y)

i

j

o x

y1 ii

1 jj

0 ji

19

Page 20: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D Cartesian coordinate system:

P: (x,y)

i

j

o x

y

1 ii

1 jj

0 ji

20

jyixop

Page 21: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

21

Page 22: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

22

Given the coordinates (x’,y’) in i’j’

- how to compute the coordinates (x,y) in ij?

Page 23: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

23

Given the coordinates (x’,y’) in i’j’

- how to compute the coordinates (x,y) in ij?

0x

0y

Page 24: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

),( 00 yx

24

Given the coordinates (x’,y’) in i’j’

- how to compute the coordinates (x,y) in ij?

1100

cossin

sincos

10

0

y

x

y

x

y

x

Page 25: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

jyixop

ji ij

),( 00 yx

25

Page 26: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

jyixop

jyixpo

ji ij

),( 00 yx

26

Page 27: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

jyixop

jyixpo

jyixoo

00

ji ij

),( 00 yx

27

Page 28: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

poooop

ji ij

),( 00 yx

28

Page 29: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

poooop

jyixop

jyixpo

jyixoo

00

ji ij

),( 00 yx

29

Page 30: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

poooop

ji ij

jyixjyixjyix

00

),( 00 yx

30

Page 31: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

poooop

ji ij

jyixjyixjyix

00

jyixjyyixx

)()( 00

),( 00 yx

31

Page 32: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

poooop

ji ij

jyixjyixjyix

00

jyixjyyixx

)()( 00

y

xji

yy

xxji

0

0),( 00 yx

32

Page 33: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

yy

xxji

0

0

),( 00 yx

33

Page 34: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

yy

xxji

0

0

y

xjiji

yy

xx 1

0

0

),( 00 yx

34

Page 35: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

yy

xxji

0

0

y

xjiji

yy

xx 1

0

0

y

xji

j

i

yy

xxT

T

0

0

),( 00 yx

35

10

01

jjij

jiii

j

iji

T

T

Page 36: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

36

Page 37: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

37

What does this column vector mean?

Page 38: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

38

What does this column vector mean? Vector i’ in the new reference system

Page 39: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

39

What does this column vector mean?

Page 40: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

40

What does this column vector mean? Vector j’ in the new reference system

Page 41: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

41

What does this column vector mean?

Page 42: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

y

xji

j

i

yy

xxT

T

0

0

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

42

What does this column vector mean? The old origin in the new reference system

Page 43: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation

pj

i

i

j

o

o

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

43

Page 44: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation

pj

i

i

j

o

o

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

?

? ?

?

44

Page 45: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation

pj

i

i

j

o

o

110010

0

y

x

yjjij

xjiii

y

xTT

TT

1

0

0

1

),( 00 yx

45

Page 46: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

?

46

Page 47: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

cos

10

0

y

x

yjjij

xji

y

xTT

T

),( 00 yx

?

47

Page 48: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

sin

cos

10

0

y

x

yjj

xji

y

xT

T

),( 00 yx

48

Page 49: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

sin

cos

10

0

y

x

yjj

xji

y

xT

T

),( 00 yx

?

49

Page 50: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

sin

sincos

10

0

y

x

yjj

x

y

xT

),( 00 yx

50

Page 51: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

sin

sincos

10

0

y

x

yjj

x

y

xT

),( 00 yx

?

51

Page 52: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o

o

1100

cossin

sincos

10

0

y

x

y

x

y

x

),( 00 yx

52

Page 53: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem

i

j

o

pj

i

o ),( 00 yx

53

Page 54: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem

i

j

o

pj

i

o ),( 00 yx

54

),()( 00 yxTR

Page 55: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem

i

j

o

p

j

io ),( 00 yx

55

),()( 00 yxTR

Page 56: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transforms the point from (x,y) to (x’,y’)

i

j

o

p

j

io ),( 00 yx

56

1100

10

01

100

0cossin

0sincos

10

0

y

x

y

x

y

x

Page 57: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transforms the point from (x’,y’) to (x,y)

i

j

o

p

j

io ),( 00 yx

57

1100

0cossin

0sincos

100

10

01

1

11

0

0

y

x

y

x

y

x

Page 58: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transform the point from (x’,y’) to (x,y)

i

j

o

p

j

io ),( 00 yx

58

1100

0cossin

0sincos

100

10

01

1

11

0

0

y

x

y

x

y

x

Page 59: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transform the point from (x’,y’) to (x,y)

i

j

o

p

j

io ),( 00 yx

59

1100

0cossin

0sincos

100

10

01

1

11

0

0

y

x

y

x

y

x

100

10

01

0

0

y

x

Page 60: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transform the point from (x’,y’) to (x,y)

i

j

o

p

j

io ),( 00 yx

60

1100

0cossin

0sincos

100

10

01

1

11

0

0

y

x

y

x

y

x

100

10

01

0

0

y

x

Page 61: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

An alternative way to look at the problem This transform the point from (x’,y’) to (x,y)

i

j

o

p

j

io ),( 00 yx

61

1100

0cossin

0sincos

100

10

01

1

11

0

0

y

x

y

x

y

x

100

10

01

0

0

y

x

100

0cossin

0sincos

Page 62: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Same results!

pj

i

i

j

o

o

1100

cossin

sincos

10

0

y

x

y

x

y

x

),( 00 yx

62

Page 63: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o o

1???

???

???

1

y

x

y

x

30

63

Page 64: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

pj

i

i

j

o o

1100

030cos30sin

030sin30cos

1

y

x

y

x

30

64

Page 65: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

p

j

i

i

j

oo

1???

???

???

1

y

x

y

x

45

65

Page 66: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

p

j

i

i

j

oo

1100

045cos45sin

045sin45cos

1

y

x

y

x

45

66

Page 67: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation&rotation

p

j

i

i

j

o

)2,4( o

1???

???

???

1

y

x

y

x

45

67

Page 68: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

2D translation & rotation

p

j

i

i

j

o

)2,4( o

1100

245cos45sin

445sin45cos

1

y

x

y

x

45

68

Page 69: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

How to model 2D movement of characters or robots?

69

Click here

Page 70: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

A 2D lamp character

1

2

3

),,( 0yx

3c

2c

1c

0c

A

70

0

Page 71: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

A 2D lamp character3l

A2l

1l

0l

71

Page 72: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

How can we draw the character given the pose ?

1

2

3

),,( 0yx

3c

2c

1c

0c

A

72

),,,,,( 3210 yx

Page 73: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

),,( 0yx

3c2c

0c

A1c

Articulated Character

A default pose (0,0,0,0,0,0)

Page 74: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the pose?

)1,5(

74

?)?,?,?,,1,5(

300

Page 75: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the pose?

)1,5(

75

)90,0,90,30,1,5(

300

Page 76: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

A 2D lamp character

1

2

3

3c

2c

1c

A

Given , and the local coordinates of point A, how to compute the global coordinates of point A?

),,,,,( 3210 yx

?

),,( 0yx0c

76

Page 77: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s local coordinate ?

1

2

3

3c

2c

1c

A

?

A

),,( 0yx0c

77

Page 78: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s local coordinate ?

1

2

3

3c

2c

1c

A

?

A

1

03

0

l

p

),,( 0yx0c

03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp 78

Page 79: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

?

),,( 0yx0c

1

00

l

p

03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp

100

010

01

)0,(2

2

l

lT

79

100

0cossin

0sincos

)( 33

33

3

R

Page 80: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

?

),,( 0yx0c

1

00

l

p

03221100 )()0,()()0,()()0,()(),( pRlTRlTRlTRyxTp

100

010

01

)0,(1

1

l

lT

80

100

0cossin

0sincos

)( 22

22

2

R

Page 81: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

?

),,( 0yx0c

1

00

l

p

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

100

10

001

),0( 00 llT

81

100

0cossin

0sincos

)( 11

11

1

R

Page 82: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

?

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

1

00

l

p

100

cossin

sincos

)(),( 00

00

0 y

x

RyxT

82

Page 83: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

83

Page 84: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the pose?

)1,5(

84

)90,0,90,30,1,5(

300

l0=1, l1=7, l2=6, l3=6

)90()0,6()0()0,7()90()1,0()30()1,5( RTRTRTRTp 0p

Page 85: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 2D Transformation

What’s the pose?

)1,5(

85

)90,0,90,30,1,5(

300

l0=1, l1=7, l2=6, l3=6

)90()0,6()0()0,7()90()1,0()30()1,5( RTRTRTRTp 0p

How about this point?

Page 86: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

How to Animate the Character?

A 2D lamp character

1

2

3

),,( 0yx

3c

2c

1c

0c

A

86

Page 87: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

How to Animate the Character?

Keyframe animation

- Manually pose the character by choosing appropriate values for

- Linearly interpolate the inbetween poses.

- Works for any types of articulated characters!

87

),,,,,( 3210 yx

Page 88: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

How to Animate the Character?

Keyframe animation

Motion capture

- motion interpolations, graph based, statistics-based,

88

Page 89: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

How to Animate the Character?

Keyframe animation

Motion capture

- motion interpolations, graph based, statistics-based, Physically based animation - needs to solve extremely complex equations

- usually very slow

- human motion might look robotic

89

Page 90: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

A 3D point (x,y,z) – x,y, and z coordinates We will still use column vectors to represent

points Homogeneous coordinates of a 3D point

(x,y,z,1) Transformation will be performed using 4x4

matrix

x

y

z

90

Page 91: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Right-handed Coordinate System

x

y

z

;zyx ;xzy yxz

Left hand coordinate system

Not used in this class and

Not in OpenGL 91/94

Page 92: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

Very similar to 2D transformation

Translation transformation

11000

100

010

001

1

z

y

x

t

t

t

z

y

x

x

y

x

Homogenous coordinates

92

Page 93: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

Very similar to 2D transformation

Scaling transformation

11000

000

000

000

1

z

y

x

s

s

s

z

y

x

z

y

x

Homogenous coordinates

93

Page 94: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

3D rotation is done around a rotation axis Fundamental rotations – rotate about x, y, or

z axes Counter-clockwise rotation is referred to as

positive rotation (when you look down negative axis)

x

y

z

+

94

Page 95: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

Rotation about z – similar to 2D rotation

x

y

z

+

11000

0100

00cossin

00sincos

1

z

y

x

z

y

x

zz

yxy

yxx

'

cossin'

sincos'

95

Page 96: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

Rotation about y: z -> y, y -> x, x->z

x

y

z

11000

0cos0sin

0010

0sin0cos

1

z

y

x

z

y

x

yy

xzx

xzz

'

cossin'

sincos'

y

z

x

96

Page 97: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Transformation

Rotation about x (z -> x, y -> z, x->y)

x

y

z

11000

0cossin0

0sincos0

0001

1

z

y

x

z

y

x

xx

zyz

zyy

'

cossin'

sincos'

z

x

y

97

Page 98: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotate p about the by the angle

x

y

zr

),,( zyxp

r

98

Page 99: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3-D Rotation General rotations in 3-D require rotating

about an arbitrary axis of rotation

Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra

99

Page 100: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotate p about the by the angle

x

y

zr

),,( zyxp

r

100

Page 101: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3-D Rotation General rotations in 3-D require rotating about

an arbitrary axis of rotation

Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra

Standard approach: express general rotation as composition of canonical rotations Rotations about x, y, z

101

Page 102: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composing Canonical Rotations Goal: rotate about arbitrary vector r by θ

Idea: we know how to rotate about x,y,zSo, rotate about z by - until r lies in the xz

planeThen rotate about y by -β until r coincides

with +zThen rotate about z by θThen reverse the rotation about y (by β )Then reverse the rotation about z (by )

102

Page 103: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotate p about the by the angle

x

y

zr

),,( zyxp

r

103

Page 104: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Translate so that rotation axis passes through the origin

x

y

zr

TRRRRRT zyzyz )()()()()(1

104

Page 105: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotation by about z-axis

x

y

z

r

TRRRRRT zyzyy )()()()()(1

105

Page 106: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotation by about y-axis

x

y

z

r

xTRRRRRT zyzyy )()()()()(1

106

Page 107: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotation by about z-axis

x

y

z

r

xTRRRRRT zyzyy )()()()()(1

107

Page 108: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotation by about y-axis

x

y

z

r

xTRRRRRT zyzyy )()()()()(1

108

Page 109: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Rotation by about z-axis

x

y

z

r

TRRRRRT zyzyz )()()()()(1

109

Page 110: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Translate the object back to original point

x

y

z

r

TRRRRRT zyzyz )()()()()(1

110

Page 111: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Final transformation matrix for rotating about an arbitrary axis

x

y

z

r

TRRRRRT zyzyz )()()()()(1

111

Page 112: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Final transformation matrix for rotating about an arbitrary axis

x

y

z

r

TRRRRRT zyzyz )()()()()(1

112

Page 113: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Final transformation matrix for rotating about an arbitrary axis

TRRRRRT zyzyz )()()()()(1

1000

0

0

0

333231

232221

131211

rrr

rrr

rrr

Page 114: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Rotation about Arbitrary Axes

Final transformation matrix for rotating about an arbitrary axis

TRRRRRT zyzyz )()()()()(1

A 3 by 3 Rotation matrix

1000

0

0

0

333231

232221

131211

rrr

rrr

rrr

Page 115: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent)normalized (columns/rows length of 1)

115

Page 116: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent)normalized (columns/rows length of 1)

The inverse of an orthogonal matrix is just its transpose:

116

T

rrr

rrr

rrr

rrr

rrr

rrr

333231

232221

131211

1

333231

232221

131211

Page 117: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent)normalized (columns/rows length of 1)

The inverse of an orthogonal matrix is just its transpose:

117

T

rrr

rrr

rrr

rrr

rrr

rrr

333231

232221

131211

1

333231

232221

131211

Page 118: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent)normalized (columns/rows length of 1)

The inverse of an orthogonal matrix is just its transpose:

118

T

rrr

rrr

rrr

rrr

rrr

rrr

333231

232221

131211

1

333231

232221

1312111r

2r

3r

Page 119: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Why?

Rotation Matrices

119

332313

322212

312111

331

1

1

1

3

2

1

3

2

1

rrrrrr

rrrrrr

rrrrrr

rrr

r

r

r

r

r

r

r

r

r

TTT

T

Page 120: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Why?

Rotation Matrices

120

332313

322212

312111

331

1

1

1

3

2

1

3

2

1

rrrrrr

rrrrrr

rrrrrr

rrr

r

r

r

r

r

r

r

r

r

TTT

T

Page 121: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Why?

Rotation Matrices

121

332313

322212

312111

331

1

1

1

3

2

1

3

2

1

rrrrrr

rrrrrr

rrrrrr

rrr

r

r

r

r

r

r

r

r

r

TTT

T

Page 122: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Why?

Rotation Matrices

122

100

010

001

331

1

1

1

3

2

1

3

2

1

TTT

T

rrr

r

r

r

r

r

r

r

r

r

Page 123: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Rotation Matrices

Orthonormal matrix: orthogonal (columns/rows linearly

independent)normalized (columns/rows length of 1)

The inverse of an orthogonal matrix is just its transpose: e.g.,

T

xx

Tyy

Tzz

RR

RR

RR

)()(

)()(

)()(

1

1

1

123

Page 124: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

),,( 000 zyx

k

'k

124

Page 125: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

2D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

110010

0

y

x

yjjij

xjiii

y

xTT

TT

),( 00 yx

poooop

125

Page 126: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

1

'

'

'

1000

'

'

'

10

0

0

z

y

x

zkkjkik

ykjjjij

xkijiii

z

y

x

TTT

TTT

TTT

),,( 000 zyx

k

'k

poooop

126

Page 127: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Coordinate Transformation

Transform object description from to

pj

i

i

j

o

o

ji ij

1

'

'

'

1000

'

'

'

10

0

0

z

y

x

zkkjkik

ykjjjij

xkijiii

z

y

x

TTT

TTT

TTT

),,( 000 zyx

k

'k

poooop

127

Page 128: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Coordinate Transformation

Transform object description from to

i

i

j

o

ji ij

1

'

'

'

1000

'

'

'

10

0

0

z

y

x

zkkjkik

ykjjjij

xkijiii

z

y

x

TTT

TTT

TTT

k

128

p'i

'j

Page 129: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

3D Coordinate Transformation

Transform object description from to

i

j

o

ji ij

k

129

p'i

'j

1

'

'

'

1000

0100

00cossin

00sincos

1

z

y

x

z

y

x

Page 130: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 3D Transformation

Similarly, we can easily extend composite transformation from 2D to 3D

1

2

3

),,( 0yx

3c

2c

1c

0c

A

130

Page 131: Jinxiang Chai Composite Transformations and Forward Kinematics 0.

Composite 3D Transformation

xRRhTRhTRRRhTRRRzyxTp )()()0,,0()()0,,0()()()()0,,0()()()(),,( 332211110000000

131