Matrix Methods in Kinematics - School of Engineeringmoreno/Kinematics/ME 5150 Kinematics...

1

Matrix Methods in Kinematics

Rigid Body Rotation Matrices

Rigid Body – points have same relative position

Displacement = Rotation and Translation

Angular rotations described about

1. Right hand Cartesian axes (x,y,z)

2. Arbitrary Axis

3. Euler Angles

Vector Methods with Matrix Notation

http://www.uconn.edu/index.php

2


Holonomic Systems Holonomic Constraints

Relations between co-ordinates and possibly time)

0),,..,( 21 tqqqf n

Rigid body

ir

jrijL

xyz displacement is not order dependent


3


Rigid body rotations

Rotations not commutative (order dependent)

z

y

x


4


1. Rotate about Z (α)

zv

yv

xv

zv

yv

xv

zvzv

vvv

vvv

yxy

yxx

1

1

1

2

2

2

12

112

112

100

0cossin

0sincos

cossin

sincos

Rotation matrix

Components in the fixed system

x-y


5


Rotation about three Cartesian axes

z

y

x

z

y

x

v

v

v

v

v

v

1

1

1

2

2

2

100

0cossin

0sincos

z

y

x

z

y

x

v

v

v

v

v

v

1

1

1

'2

'2

'2

cos0sin

010

sin0cos

z

y

x

z

y

x

v

v

v

v

v

v

1

1

1

"2

"2

"2

cossin0

sincos0

001

y

z

x

X,Y,Z axes fixed in space

'2v 2v

''2v

1v


6


Plane rotation (2D) (rotation about z)

Spatial Rotation – sequence z,y,x

1,2 vRv z

1,,1,,,2 vRvRRRv zyx

100

0cossin

0sincos

cos0sin

010

sin0cos

cossin0

sincos0

001


7


1,,1,,,2 vRvRRRv zyx

12 v

CCCSSSCCSCSS

SCSSSCCSSCCS

SCSCC

v

X

Y

Z

dependentsequence

axesfixedabout

.3

.2

.1

etcSC ,sin,cos


8


2. Rotation about axis u (φ)

• Decompose to rotations about x, y, z

• Rotate u about z and then back again

+/- show directions

x

y

z

u

1,1,,,,,2 vRvRRRRRv uyxzxy

xu

yu

zu

cos0sin

010

sin0cos

cossin0

sincos0

001

1v


9


y

),,,,,(2

1,1,,,,,2

cscscsfv

vRvRRRRRv uyxzxy

z

x

zx

zzx

zx

xy

u

u

uu

uuu

uu

uu

coscos

sincos

coscos

sinsin

22

22

22

x

z

u

xu

yu

zu

1v

22

zx uu


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1,1,,,,,2 vRvRRRRRv uyxzxy

1

2

2

2

2 v

CVuSuVuuSuVuu

SuVuuCVuSuVuu

SuVuuSuVuuCVu

v

zxzyyzx

xzyyzyx

yzxzyxx

cos,sin

cos1

),,,(,

CS

V

uuufR zyxu

spaceinfixedu

ofdiruuu zyx

)(

cos,,


11


3. Euler Angles

Displacement of a rigid body in terms

of three relative displacement angles

Each rotation about an axis whose

location depends on preceding

rotation

anglespin

anglenutation

angleprecision

rotationzxz

"

'"","'','

'',',,

zabout

xaboutthenzyxzyx

zaboutbyzyxzyx

3

2

1

• 4 sets of axes initially coincident

• xyz fixed in the body


12



13



14


z-x-z

x-y-x

y-z-y

z-y-z

x-z-x

y-x-y

x-y-z

y-z-x

z-x-y

x-z-y

z-y-x

y-x-z

12 different possible rotation sets


15


zxz

z

T

zxz

T

z

T

xzxz

T

z

T

xzxzz

T

zxzx

zxz

RRRR

RRRRRRRRRR

RRRRRR

RRRR

RRRR

,,,,,

,,,,,,,,,,,

,,,,,",

,,,',

,',",,,

basesdifferentoperatorsame

similarareAandB

APPB

tiontransformasimilarity

,

,

1


16


CCSSS

SCCCCSSSCCCS

SSCCSSCSCSCC

R

getcaseeitherRRRR

axesfixedinitialabouttiontransformasimilaritywithor

RRRR

zxz

zxz

,,

,,,,,

,',",,,

:

)(

etcR z

100

0cossin

0sincos

,


17


Euler Angles

3

2

1

1

2

3

Angular Velocity Vectors

Rigid

body


18


Rotation of Rigid Body in 2d Cartesian Space

ppRqRqalso

ppqRq

pqRpq

pq

pq

pq

pq

yy

xx

yy

xx

11

11

11

11

11

cossin

sincos

p1 q1

Fixed x-y

Vector form

specifiedqwhenqfind

knownandppif

1

1,


19


1100

)cossin(cossin

)sincos(sincos

1

cossincossin

sincossincos

cossin

sincos

cossin

sincos

1

1

1

11

11

1111

1111

1

1

1

1

z

y

x

yxy

yxx

z

y

x

yyxyxy

xyxyxx

y

x

y

x

y

x

y

x

q

q

q

ppp

ppp

q

q

q

pppqqq

pppqqq

p

p

p

p

q

q

q

q

1 2


20


1100

)cossin(cossin

)sincos(sincos

1

1

1

11

11

y

x

yxy

yxx

y

x

q

q

ppp

ppp

q

q

11

1100

)(

1

1

1

1

1

qD

q

q

qpRpR

q

q

y

x

y

x

3x3 D = plane displacement matrix


21


Spatial (3D) Rigid Body Displacement

Replace

ppqRq

pqRpq

RorRorRwithR u

11,,

11,,

,,,,,

1100011

11,,,,1,,

qpRpRqD

q

Cartesian u Axis Euler

4x4 matrix (3D) new original

Using Cartesian


22


1

1000

(

1

)(

1

1

1

1,1,

11,1

11

1

,

z

y

x

uu

u

u

q

q

q

pRsupR

qz

qy

qx

pqRsupq

pqRpq

suppreplace

uaxisalongrotationRuse

Screw displacement matrix

Screw displacement matrix

p1

u

s

p=p1+su

q

q1

x

z

y


23


1100011

11,,,,1,,

qpRpRqD

q

12 non constant elements

3D space= 6 DOF, 6 elements are dependent

To Define D: euler angles, dir cos, points on body, etc…


24


Example Displacement of a point

Moving with a rigid body

2112 ppRqRq

732.3

0.4

2

3

11

13

60cos60sin

60sin60cos

60cos60sin

60sin60cos

2

2

2

2

2

2

11

11

2

2

y

x

y

x

y

x

yy

xx

y

x

q

q

q

q

p

p

pq

pq

q

q


25

1

732.3

0.4

1

1

3

100

634.05.0866.

366.3866.5.0

1

1100

)60cos160sin12(60cos60sin

)60sin160cos13(60sin60cos

1

1100

)cossin(cossin

)sincos(sincos

1

2

2

1

1

2

2

1

1

112

112

2

2

2

y

x

y

x

y

x

y

x

yxy

yxx

z

y

x

q

q

q

q

q

q

q

q

ppp

ppp

q

q

q

3x3 Displacement matrix



26


Finite Rotation Pole – plane rotation about p0

With new position vectors

p1=p2=p0

po


27


Displacement matrix D now written as:

1

732.3

0.4

1

1

3

100

634.05.0866.

366.3866.5.0

1100

)cossin(cossin

)sincos(sincos

1

1

1

000

000

2

2

y

x

yxy

yxx

y

x

q

q

ppp

ppp

q

q

Previously

p1 and p2 in

q2 is the

same point Original

displacement

matrix


28


634.0

366.3

cos1sin

sincos1

634.0cossin

366.3sincos

0

0

1212

1212

000

000

y

x

yxy

yxx

p

p

ppp

ppp

232.3

134.1

0

0

y

x

p

p


29


HW #5 Salute

Z

Y

X

shoulder

p1=elbow

q1=tip of finger

p

q

30◦

Use Cartesian angles to find [D]

Treat as 3 independent rotations


30


HW #5 Salute

Z

Y

X

shoulder

p1=elbow

q1=tip of finger

p

q

30◦

z-y-z rotation

Z

Y

X

180

Z

X

Changed axes

notation

30

3

2

1

Y move origins for

2,3 rotations


31


Finding the Displacement Matrix by Inversion

x

y

C1

A1

B1

B2

C2

A2 1,2,61,5,1

1,1,71,6,2

1,1,51,4,2

21

21

21

CC

BB

AA

Known points


32


Displacement Matrix by Inversion

1

1

1

1

1

1

1

333231

232221

131211

1

11

111

y

x

y

x

y

x

y

x

y

x

q

q

q

q

D

q

q

Dq

q

aaa

aaa

aaa

q

q

qDq


33


1

111

564

122

111

211

675

111

564

122

111

211

675

D

D

d a

D=d*a-1

2D Planar motion

x

y

z

q q1


34


ij

ji

ij M

AAA

AAA

AAA

AadjA

AadjA

A

)1(

333231

232221

131211

332313

322212

312111

1

ith row=ith column

α=co factor

Mij= minor of A

Finding the inverse of A (by hand – the long way)

adj - adjoint


35


d =

5 7 6

1 1 2

1 1 1

0 1 1

-1 0 3

0 0 1

100

)cossin(cossin

)sincos(sincos

112

112

yxy

yxx

ppp

ppp

a =

2 2 1

4 6 5

1 1 1

inv(a)

ans =

0.5000 -0.5000 2.0000

0.5000 0.5000 -3.0000

-1.0000 0 2.0000

90

Using MATLAB

Displacement matrix

D=d*a-1


36


132.1725

11032.2

1015

3

3

3

c

b

a

12015

1035

1015

2

2

2

c

b

a

11520

1350

1150

1

1

1

c

b

a

X

Y

1a

2b

2c

3a2a

3c

3b

1b

1c

30

HW 5 by Inversion

Using My arm


37


By Inverse method – position 1 2

111

153515

2000

111

2000

153515

12D

2a 2b 2c 1a 1b 1c

-0.0500 0.0500 1.7500

0 -0.0500 -0.7500

0.0500 0 0

D =

0 1.0000 0

-1.0000 0 0

0 0 1.0000

inv

12D

90


38


By Inverse method – position 2 3

111

2000

153515

111

32.17100

2532.215

23D

3a3b 3c

2a 2b 2c

0.0500 0.0500 1.7500

-0.0500 0 -0.7500

0 -0.0500 0

inv

23D -0.8660 0.5000 -27.9900

-0.5000 -0.8660 -7.5000

0 0 1.0000

150


39


1

10

32.2

1

35

0

100

5.75.866.

99.27866.5.

133

122313

Db

DDD240


Matrix Methods in Kinematics - School of Engineeringmoreno/Kinematics/ME 5150 Kinematics...

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Transcript of Matrix Methods in Kinematics - School of Engineeringmoreno/Kinematics/ME 5150 Kinematics...