Kinematics Primer

Jyun-Ming Chen

Contents

General Properties of Transform2D and 3D Rigid Body Transforms Representation Computation Conversion …

Transforms for Hierarchical Objects

Math Primer

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

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

Ex: 2D translation

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)

Ex: 2D translation (cont)

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)

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

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)

2D Kinematics

Rigid body transform only consists of Tr(x,y) Rot(z,)

Computation: 3x3 matrix is

sufficient)Rot(z, ),Tr(

100

cossin

sincos

T

general,In

100

0cossin

0sincos

)Rot(z,

100

10

01

),Tr(

yxy

x

y

x

yx

ttt

t

t

t

tt

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

3D Rotation Representations

Axis-angle3X3 rotation matrixUnit quaternion

Learning Objectives Representation Perform rotation Composition Interpolation Conversion among

representations …

Axis-Angle Representation

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

Perform rotation Rodrigues formula

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

Rodrigues Formula

v’=R v

r

v

v’

Rodrigues (cont)

http://mesh.caltech.edu/ee148/notes/rotations.pdfhttp://www.cs.berkeley.edu/~ug/slide/pipeline/assignments/as5/rotation.html

Rotation Matrix

Meaning of three columnsPerform rotation: linear algebraComposition: trivial orthogonalization

might be required due to FP errors

Interpolation: ?

Ax

uAxuAxuAx

uxuxuxx

uuuaA ij

ˆˆˆ

ˆˆˆ

ˆˆˆ

332211

332211

321

xRRxRRxRx

xRx

12122

1

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!

Quaternion

A mathematical entity invented by HamiltonDefinition

jikki

ikjjk

kjiij

kji

qqkqjqiqqq

1222

03210

i

j k

Quaternion (cont)

Operators Addition

Multiplication

Conjugate

Length

kqpjqpiqpqpqp

kqjqiqqq

kpjpippp

33221100

3210

3210

qpqppqqpqppq 0000

qqq 0

* *** pqpq

23

22

21

20

* qqqqqqq

Unit Quaternion

Define unit quaternion as follows to represent rotation

Example Rot(z,90°)

nqn ˆsincos),ˆ(Rot 22 1q

22

22 00q

Why “unit”?

DOF point of view!

Unit Quaternion (cont)

Perform Rotation

Composition

Interpolation

)(22)(

...

020

*

xqqxqqxqqq

qxqx

****

*

)()( qpxqpqpxpqqxqx

pxpx

)(

)(

)1()( 21

tp

tpp

tppttp

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)

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

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’

:angle-axis ingCorrespond

)(

:normalize

)0( :middle

:ionInterpolat

)(

)00()00(

00

00

)00()00(

)00)(00(

:nCompositio

5.1

5.1

42

42

22

221

121

5.1

21

21

21

21

22

22

22

22

22

22

22

22

22

22

22

22

22

22

12

q

qq

qqq

kji

qq

32

221

21

61

61

32

23

42

42

22

5.1

5.1

42

42

22

221

121

5.1

cos),0(ˆ

:angle-axis ingCorrespond

)0()0(

:normalize

)0( :middle

:ionInterpolat

n

q

qq

qqq

Spatial Displacement

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

Quaternion

TxxdR

Tz

y

x

x

dRxx

10

,

1

dqxqx *

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

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

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)

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.

Ex: Two-Link Arm (VRML syntax)

Transform { rotation 0 0 1 45 children Link1 Transform { translation 0 0 6 children Transform { rotation 0 0 1 30 children Link2 } } }

Classes in Javax.vecmath

• Conversion Methods:

Exercises

Study the references of Rodrigues formulaVerify equivalence of these 2 ref’sCompute inverse Rodrigues formula