1

Kinematics ( 運動學 ) Primer

Jyun-Ming ChenFall 2009

2

Contents

General Properties of Transform2D and 3D Rigid Body Transforms DOF (degree of freedom) Representation Computation Conversion …

Transforms for Hierarchical Objects

3

Kinematic Modeling

Two interpretations of transform “Global”:

An operator that “displaces” a point (or set of points) to desired location

“Local”: specify where

objects are placed in WCS by moving the local frame

Explain these concepts via 2D translationVerify that the same holds for rotation, 3D, …

4

Ex: 2D translation (move point)

x

y

p

The transform, as an operator, takesp to p’, thus changing the coordinate of p:

Tr(t) p = p’

p’

1

4

4

1

22

31

1

2

1

100

210

301

Tr(t)

5

Ex: 2D translation (move frame)

p’

x

y

x’

y’

p

The transform moves the xy-frame tox’y’-frame and the point is placedwith the same local coordinate.

To determine the corresponding position of p’ in xy-frame:

frame y' x' toframe- xys that take transform the:

y'-framein x' of coordinate the:

frame-in xy of coordinate the:

''

''

''''

T

pp

pp

ppT

xyyx

yx

xy

xyyxxyyx

1

4

4

1

22

31

1

2

1

100

210

301

Tr(t)

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Properties of Transform

Transforms are usually not commutable TaTb p TbTa p (in

general)

Rigid body transform: the ones

preserving the shape

Two types: rotation rot(n,) translation tr(t)

Rotation axis n passes thru

origin

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Rigid Body Transform

transforming a point/object rot(n,) p; tr(t) p

not commutable rot(n,) tr(t) p tr(t) rot(n,) p

two interpretations (local vs. global axes, see next pages)

8

Hierarchical Objects

For modeling articulated objects Robots, mechanism, …

Goals: Draw it Given the configuration, able to

compute the (global) coordinate of every point on body

9

Ex: Two-Link Arm (2D)

Configuration Link 1: Box (6,1);

bend 45 deg Link 2: Box (8,1);

bend 30 deg

Goals: Draw it find tip position

x

y

x

y

10

Ex: Two-Link Arm

Tr(0,6) Rot(z,45)Rot(z,30)

Tip pos:(0,8)

8

0Rot(z,30)Tr(0,6)Rot(z,45)

Tip Position:

T for link1: Rot(z,45) Tr(0,6) Rot(z,30)

T for link2: Rot(z,45)

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Ex: Two-Link Arm

Rot(z,45)

x’

y’

Tr(0,6’)

x”

y”

Rot(z”,30)Tip pos:(0’”,8’”)

x”’

y’”

8

0Rot(z,45))Tr(0,6',30)Rot(z"

),T(),T(),T(

),T()','T()","T(

shown that becan It

112233

332211

ppp

ppp

Thus, two views are equivalent

The latter might be easier tovisualize.

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2D KinematicsRigid body transform only consists of Tr(x,y) Rot(z,)

Computation: 3x3 matrix is

sufficient to realize Tr and Rot

)Rot(z, ),Tr(

100

cossin

sincos

T

general,In

yxy

x

ttt

t

100

0cossin

0sincos

)Rot(z,

100

10

01

),Tr(

y

x

yx t

t

tt

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3D Kinematics

Consists of two parts 3D rotation 3D translation

The same as 2D

3D rotation is more complicated than 2D rotation (restricted to z-axis)Next, we will discuss the treatment for spatial (3D) rotation

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DOF (degree of freedom)

… of a system of moving bodies is the number of independent variables required to specify the configuration… is closely related to kinematic representation(# of independent variables) = (# of total variables) ̶ (# of equality constraints)

15

DOF of …

A particle in R2

R3

A rigid body inR2

R3

2

3

3

2

4 (Plucker coordinates)

a line in R2 a line in R3

a plane in R3

3

6

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3D Rotation Representations

Euler anglesAxis-angle3X3 rotation matrixUnit quaternion

Learning Objectives Representation

(uniqueness) Perform rotation Composition Interpolation Conversion among

representations …

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Euler AnglesRoll, pitch, yaw

Ref: http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf

Gimbal lock: reduced DOF due to overlapping axes

Weird Interpolation …

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Previous[-26, -90, 127]

Next[71, 68, -105]

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Axis-Angle Representation

Rot(n,) n: rotation axis (global) : rotation angle (rad. or deg.) follow right-handed rule

Rot(n,)=Rot (-n,-)Problem with null rotation: rot(n,0), any nPerform rotation Rodrigues formula (next page)

Interpolation/Composition: poor Rot(n2,2)Rot(n1,1) =?= Rot(n3,3)

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Rodrigues Formula

v’=R v

r

v

v’

References:

http://mesh.caltech.edu/ee148/notes/rotations.pdf

http://www.cs.berkeley.edu/~ug/slide/pipeline/assignments/as5/rotation.html

r : unit vector (axis)

This is the “cross-product matrix”r v = r v

This is the “cross-product matrix”r v = r v

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Rotation Matrix (xy plane)

z

100

0cossin

0sincos

),(

100

0cossin

0sincos

zRot

z

y

x

z

y

x

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Rotation Matrix

About y axis

Rodrigues formulaAbout any axis

x

y,y’

zz’

x’

cos0sin

010

sin0cos

),(yRot

Supplemental (rotation matrix & axis-angle)

Rot (a, ) Q Rot (a, ) Q

Q Rot (a, )Q Rot (a, )

Ref: url (local)

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Rotation Matrix

Three columns of R: the transformed basesPerform rotation:

x’ = Rx

Rx

uRxuRxuRx

uxuxuxx

uuuaR ij

ˆˆˆ

ˆˆˆ

ˆˆˆ

332211

332211

321

332211 ˆˆˆ uxuxuxx

x

y,y’

zz’

x’

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Rotation Matrix (example)

x

y

z y’

z’

x’

100

010

001

R

x

y

z

x’

y’

z’

100

001

010

R

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Rotation Matrix (cont)

Composition: trivial orthogonalization

might be required due to FP errors

Interpolation: ?

xRRxRRxRx

xRx

12122

1

27

Gram-Schmidt Orthogonalization

If 3x3 rotation matrix no longer orthonormal, metric properties might change!

321321 ˆˆˆˆˆˆ vvvuuu

222

231

11

1333

111

1222

11

ˆˆˆ

ˆˆˆ

ˆˆ

ˆˆˆˆ

ˆˆˆ

ˆˆˆˆ

ˆˆ

vvv

vuv

vv

vuuv

vvv

vuuv

uv

Verify!

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Transformation Matricesv = Huv = Hu

1000

000

000

000

),,(z

y

x

zyx s

s

s

sssScale

Translation Rotation (about principal axes)

Scaling

111000

000

000

000

1

zs

ys

xs

z

y

x

s

s

s

z

y

x

z

y

x

z

y

x

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Translation (math)

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Frame

1010

0

10RotTrans

Translate , Rotate

tRRtI

thenfirst

y

z

x

tx

y

z

R

Object coordinates

Recall how we did the spotlight beam!

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Quaternion

A mathematical entity invented by HamiltonDefinition

jikki

ikjjk

kjiij

kji

qqkqjqiqqq

1222

03210

i

j k

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Quaternion (cont)

Operators Addition

Multiplication

Conjugate

Length

kqpjqpiqpqpqp

kqjqiqqq

kpjpippp

33221100

3210

3210

qpqppqqpqppq 0000

qqq 0

* *** pqpq

23

22

21

20

* qqqqqqq

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Unit Quaternion

Define unit quaternion as follows to represent rotation

Example Rot(z,90°)

q and –q represent the same rotation Null rotation: q = [1,0,0,0]

nqn ˆsincos),ˆ(Rot 22 1q

22

22 00q

Why “unit”?

DOF point of view!

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Example

x

y

z

x

y

z

22

22 00q

100

001

010

Rot (90, 0,0,1) OR Rot (-90,0,0,-1)

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Unit Quaternion (cont)

Perform Rotation

Composition

Interpolation Linear Spherical linear (slerp, more later)

)(22)(

...

020

*

xqqxqqxqqq

qxqx

****

*

)()( qpxqpqpxpqqxqx

pxpx

)(

)(,)1()( 21

tp

tpptppttp

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Example

x

y,x’

z,z’

y’

1

2

1

1

1

2

100

001

010

100

001

010

Rpp

R

Rot(z,90°)Rot(z,90°)

p(2,1,1)

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Example (cont)

1

2

1

1

0

0

0

2

1

)(0

0

2

112

002

1

1

2

)(

)(22)(

22

22

22

22

21

21

020

kji

pqqpqqpqqqp

2

222 00

1120

q

p

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Example

x

y,x’

z,z’

y’

)00()001(sincos

)00()100(sincos

22

22

290

290

2

22

22

290

290

1

q

q

x,x’

y

z,y’z’

)(

)00()00(

00

00

)00()00(

)00)(00(

:nCompositio

21

21

21

21

22

22

22

22

22

22

22

22

22

22

22

22

22

22

12

kji

qq

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Matrix Conversion

23

22

21

2010322031

103223

22

21

203021

2031302123

22

21

20

)(2)(2

)(2)(2

)(2)(2

qqqqqqqqqqqq

qqqqqqqqqqqq

qqqqqqqqqqqq

R

100

001

010

00 22

22

3210

R

qqqqq

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Matrix Conversion (cont)

33221123

33221122

33221121

33221120

14

1

14

1

14

1

14

1

rrrq

rrrq

rrrq

rrrq

32234

132

311341

31

211241

21

122141

30

311341

20

233241

10

rrqq

rrqq

rrqq

rrqq

rrqq

rrqq

Find largest qi2; solve the rest

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Spherical Linear Interpolation

33221100cos

sin

sin

sin

)1(sin)(

rqrqrqrq

rt

qt

ts

The computed rotation quaternion rotates about a fixed axis at constant speed

References:http://www.gamedev.net/reference/articles/article1095.asphttp://www.diku.dk/research-groups/image/teaching/Studentprojects/Quaternion/http://www.sjbrown.co.uk/quaternions.htmlhttp://www.theory.org/software/qfa/writeup/node12.html

q

r

unit sphere in R4

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Spatial Displacement

Any displacement can be decomposed into a rotation followed by a translationMatrix

Quaternion

TxxdR

Tz

y

x

x

dRxx

10

,

1

dqxqx *