KINEMATICS POSE (POSITION AND ORIENTATION) OF A RIGID BODY

University of Bridgeport

1

Introduction to ROBOTICS

Representing Position (2D)

x

y

p

5

2

25

p (“column” vector)

yxp ˆ2ˆ5

x A vector of length one pointing in the direction of the base frame x axis

A vector of length one pointing in the direction of the base frame y axis

y 2

Representing Position: vectors• The prefix superscript denotes the

reference frame in which the vector should be understood

48

23

25

pb 32

a p

Same point, two different reference frames

p

5

2

b

ˆax

bx

byˆay

3

2

3

Representing Position: vectors (3D)

x

y

z

right-handed coordinate frame

x

y

z

252

p

p

4 zyxp ˆ2ˆ5ˆ2 x A vector of length one pointing in the

direction of the base frame x axis

A vector of length one pointing in the direction of the base frame y axis

y

A vector of length one pointing in the direction of the base frame z axis

z

pRp BB

AA

The rotation matrix

cossinsincos

BAR

cossinsincos1

BA

AB RR

ˆAx

ˆAy

ˆBx

ˆBy

pRp AA

BB

:To specify the coordinate vectors for the fame B with respect to frame A

BAR

p

5ˆAx ˆBxθ: The angle between and

in anti clockwise direction

USEFUL FORMULAS

6

1)(.

)()(1

1

RDetIRR

IRR

RRR

AB

BA

TAB

AB

BA

1010

pBExample 1

pRp BB

AA

30cos30sin30sin30cos

BAR

30

30

xA ˆ

yA ˆ

xB ˆ

yB ˆ

7

pfind A

23

21

21

23

BAR

13.66033.6603

1010

23

21

21

23

PA

BASIC ROTATION MATRIX Rotation about x-axis with

8

CSSCxRot

00

001),(

x

z

y

v

wP

u

cossin

sincos

),(

wvz

wvy

ux

w

v

u

z

y

x

ppp

ppppp

ppp

xRppp

BASIC ROTATION MATRICES Rotation about x-axis with

Rotation about y-axis with

Rotation about z-axis with

9uvwxyz RPP

CSSCxRotRx

00

001),(,

0

0100

),(,

CS

SCyRotRy

10000

),( ,

CSSC

zRotRz

EXAMPLE 2 A point is attached to a rotating

frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

10

)2,3,4(uvwp

2964.4598.0

234

10005.0866.00866.05.0

)60,( uvwxyz pzRotp

EXAMPLE 3 A point is the coordinate w.r.t.

the reference coordinate system, find the corresponding point w.r.t. the rotated OUVW coordinate system if it has been rotated 60 degree about OZ axis.

)2,3,4(xyza

uvwa

11

1

( ,60)

: ( ,60)

: ( , 60)

0.5 0.866 0 4 4.5980.866 0.5 0 3 1.964

0 0 1 2 2

Tuvw xyz

uvw xyz

uvw xyz

p Rot z p

OR p Rot z p

OR p Rot z p

COMPOSITE ROTATION MATRIX A sequence of finite rotations

rules: if rotating coordinate OUVW is rotating about principal axis

of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix [rotation about fixed frame]

if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix [rotation about current frame]

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ROTATION WITH RESPECT TO CURRENT FRAME

BB

AA PRP

13

CC

BB PRP

DD

CC PRP

DD

CC

BB

ADD

AA PRRRPRP

DC

CB

BA

DA RRRR

EXAMPLE 4 Find the rotation matrix for the following

operations:

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Post-multiply if rotate about the current frame

Pre-multiply if rotate about the fixed frame

Rotation about Y axisRotation about current Z axisRotation about current X axis

SSSCCSCCSSCSSCCCS

CSSSCCSCSSCC

CSSCCS

SCxRotzRotyRotR

00

001

10000

C0S-010

S0C),(),(),(

EXAMPLE 5 Find the rotation matrix for the following

operations:

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axis Y fixedabout Rotation axis Zfixedabout Rotation

axis Xabout Rotation

SSSCCSCCSSCSSCCCS

CSSSCCSCSSCC

CSSCCS

SCxRotzRotyRotR

00

001

10000

C0S-010

S0C),(),(),(

Pre-multiply if rotate about the fixed framePost-multiply if rotate about the current frame

EXAMPLE 6 Find the rotation matrix for the following

operations:

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axis X fixedabout Rotation axis Ycurrent about Rotation

axis Zfixedabout Rotation axis current Zabout Rotation

axis Xabout Rotation

Pre-multiply if rotate about the fixed framePost-multiply if rotate about the current frame

EXAMPLE 6 Find the rotation matrix for the following

operations:

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axis X fixedabout Rotation axis Ycurrent about Rotation

axis Zfixedabout Rotation axis current Zabout Rotation

axis Xabout Rotation

),(),(),(),(),( yRotzRotxRotzRotxRotR

QUIZ Description of Roll Pitch Yaw Find the rotation matrix for the following

operations:

18X

Y

Z

axis{YAW} Zfixedabout Rotation }axis{PITCH Y fixedabout Rotation

axis{ROLL} Xabout Rotation

ANSWER

19

X

Y

Z

axis Zfixedabout Rotation axis Y fixedabout Rotation

axis Xabout Rotation

CCSCSCSSSCSCSSCCSSSCSCSSCSSCC

CSSCCS

SCxRotyRotzRotR

00

001

C0S-010

S0C

10000

),(),(),(

HOMOGENEOUS TRANSFORMATION Special cases

1. Translation

2. Rotation

20

100

31

13BA

BA RH

10 31

33AB

BA pIH

EXAMPLE 7 Translation along Z-axis with h:

21

1000100

00100001

),(h

hzTrans

111000100

00100001

11

1

1

1

1

1

hppp

ppp

hzyx

z

y

x

z

y

x

x

y

z P

xB

yBzB

O

hx

y

zP

u

vw

O

EXAMPLE 7 Translation along Z-axis with h:

22

111000100

00100001

1hp

pp

ppp

hzyx

B

B

B

B

B

B

z

y

x

y

y

x

PHP BB

00

EXAMPLE 8 Rotation about the X-axis by

1100000000001

11

1

1

z

y

x

ppp

CSSC

zyx

23

100000000001

),(

CSSC

xRot

x

z

y

1y1z

P

1x

HOMOGENEOUS TRANSFORMATION Composite Homogeneous Transformation

Matrix Rules:

Transformation (rotation/translation) w.r.t fixed frame, using pre-multiplication

Transformation (rotation/translation) w.r.t current frame, using post-multiplication

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EXAMPLE 9 Find the homogeneous transformation

matrix (H) for the following operations:

,,,, xaxdzz RotTransTransRotH

25

:

axis OZabout ofRotation axis OZ along d ofn Translatioaxis OX along a ofn Translatio

axis OXabout Rotation

Answer

100000000001

100001000010

001

1000100

00100001

100001000000

CSSC

a

dCSSC

Remember those double-angle formulas…

sincoscossinsin

sinsincoscoscos

26

333231

232221

131211

aaaaaaaaa

A

332313

322212

312111

aaaaaaaaa

TA

333231

232221

131211

aaaaaaaaa

25

p 25Tp

Review of matrix transpose

TTT BABA Important property:

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2221

1211

aaaa

A

2221

1211

bbbb

B

and matrix multiplication…

2222122121221121

2212121121121111

2221

1211

2221

1211

babababababababa

bbbb

aaaa

AB

babb

aabababa T

y

xyxyyxx

Can represent dot product as a matrix multiply:

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HW Problems 2.10, 2.11, 2.12, 2.13, 2.14 ,2.15,

2.37, and 2.39

Quiz next class

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