Kinematics of Robot

Manipulator

Examples of Kinematics Calculations

• Forward kinematics

Given joint variables

End-effector position and orientation,

-Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyxY x

y

z

Examples of Inverse Kinematics• Inverse kinematicsEnd effector position and orientation

Joint variables -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyx

x

y

z

Example 1: one rotational link

0x

0y

1x1y

)/(cos

kinematics Inverse

sin

cos

kinematics Forward

01

0

0

lx

ly

lx

l

Robot Reference Frames– World frame– Joint frame– Tool frame

x

yz

x

z

y

W R

PT

Coordinate Transformation– Reference coordinate frame OXYZ– Body-attached frame O’uvw

wvu kji wvuuvw pppP

zyx kji zyxxyz pppP

x

y

z

P

u

vw

O, O’

Point represented in OXYZ:

zwyvxu pppppp

Tzyxxyz pppP ],,[

Point represented in O’uvw:

Two frames coincide ==>

Properties: Dot Product

• Mutually perpendicular • Unit vectors

Properties of orthonormal coordinate frame

0

0

0

jk

ki

ji

1||

1||

1||

k

j

i

Let and be arbitrary vectors in and be the angle from to , then

3R

cosyxyx

x yx y

Coordinate Transformation

• Coordinate Transformation– Rotation only

wvu kji wvuuvw pppP

x

y

zP

zyx kji zyxxyz pppP

uvwxyz RPP u

vw

How to relate the coordinate in these two frames?

Basic Rotation• Basic Rotation

– , , and represent the projections of onto OX, OY, OZ axes, respectively

– Since

xp Pyp zp

wvux pppPp wxvxuxx kijiiii

wvuy pppPp wyvyuyy kjjjijj

wvuz pppPp wzvzuzz kkjkikk

wvu kji wvu pppP

Basic Rotation Matrix• Basic Rotation Matrix

– Rotation about x-axis with

w

v

u

z

y

x

p

p

p

p

p

p

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

x

z

y

v

wP

u

CS

SCxRot

0

0

001

),(

Is it True? Can we check?

• – Rotation about x axis with

cossin

sincos

cossin0

sincos0

001

wvz

wvy

ux

w

v

u

z

y

x

ppp

ppp

pp

p

p

p

p

p

p

x

z

y

v

wP

u

Basic Rotation Matrices– Rotation about x-axis with

– Rotation about y-axis with

– Rotation about z-axis with

uvwxyz RPP

CS

SCxRot

0

0

001

),(

0

010

0

),(

CS

SC

yRot

100

0

0

),(

CS

SC

zRot

Matrix notation for rotations

• Basic Rotation Matrix

– Obtain the coordinate of from the coordinate of

z

y

x

w

v

u

p

p

p

p

p

p

zwywxw

zvyvxv

zuyuxu

kkjkik

kjjjij

kijiii

uvwxyz RPP

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

R

xyzuvw QPP

TRRQ 1

31 IRRRRQR T

uvwP

xyzP

<== 3X3 identity matrix

Dot products are commutative!

Example 2: rotation of a point in a rotating frame

• 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.

)2,3,4(uvwa

2

964.4

598.0

2

3

4

100

05.0866.0

0866.05.0

)60,( uvwxyz azRota

Example 3: find a point of rotated system for point in coordinate system

• A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.

)2,3,4(xyza

uvwa

2

964.1

598.4

2

3

4

100

05.0866.0

0866.05.0

)60,( xyzT

uvw azRota

• Reference coordinate frame OXYZ• Body-attached frame O’uvw

Composite Rotation Matrix

• A sequence of finite rotations – matrix multiplications do not commute– rules:

• if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

• 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

Example 4: finding a rotation matrix

• Find the rotation matrix for the following operations:

Post-multiply if rotate about the OUVW axes

Pre-multiply if rotate about the OXYZ axes

...

axis OUabout Rotation

axisOW about Rotation

axis OYabout Rotation

Answer

SSSCCSCCSSCS

SCCCS

CSSSCCSCSSCC

CS

SCCS

SC

uRotwRotIyRotR

0

0

001

100

0

0

C0S-

010

S0C

),(),(),( 3

Reference coordinate frame OXYZBody-attached frame O’uvw

Coordinate Transformations• position vector of P

in {B} is transformed to position vector of P in {A}

• description of {B} as seen from an observer in {A}

Rotation of {B} with respect to {A}

Translation of the origin of {B} with respect to origin of {A}

Coordinate Transformations• Two Special Cases

1. Translation only– Axes of {B} and {A} are

parallel

2. Rotation only– Origins of {B} and {A} are

coincident

1BAR

'oAPBB

APA rrRr

0' oAr

Homogeneous Representation• Coordinate transformation from {B} to {A}

• Homogeneous transformation matrix

'oAPBB

APA rrRr

1101 31

' PBoAB

APA rrRr

10101333

31

' PRrRT

oAB

A

BA

Position vector

Rotation matrix

Scaling

Homogeneous Transformation• Special cases

1. Translation

2. Rotation

10

0

31

13BA

BA RT

10 31

'33

oA

BA rIT

Example 5: translation• Translation along Z-axis with h:

1000

100

0010

0001

),(h

hzTrans

111000

100

0010

0001

1

hp

p

p

p

p

p

hz

y

x

w

v

u

w

v

u

x

y

z P

u

vw

O, O’hx

y

z

P

u

vw

O, O’

Example 6: rotation• Rotation about the X-axis by

11000

00

00

0001

1w

v

u

p

p

p

CS

SC

z

y

x

1000

00

00

0001

),(

CS

SCxRot

x

z

y

v

wP

u

Homogeneous Transformation

• Composite Homogeneous Transformation Matrix

• Rules:– Transformation (rotation/translation) w.r.t (X,Y,Z)

(OLD FRAME), using pre-multiplication

– Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication

Example 7: homogeneous transformation

• Find the homogeneous transformation matrix (T) for the following operations:

44,,,, ITTTTT xaxdzz

:

axis OZabout ofRotation

axis OZ along d ofn Translatio

axis OX along a ofn Translatio

axis OXabout Rotation

Answer

1000

00

00

0001

1000

0100

0010

001

1000

100

0010

0001

1000

0100

00

00

CS

SC

a

d

CS

SC

Homogeneous Representation• A frame in space (Geometric

Interpretation)

x

y

z),,( zyx pppP

1000zzzz

yyyy

xxxx

pasn

pasn

pasn

F

n

sa

101333 PR

F

Principal axis n w.r.t. the reference coordinate system

(X’)

(y’)(z’)

Homogeneous Transformation

• Translation

y

z

n

sa n

sa

1000

10001000

100

010

001

zzzzz

yyyyy

xxxxx

zzzz

yyyy

xxxx

z

y

x

new

dpasn

dpasn

dpasn

pasn

pasn

pasn

d

d

d

F

oldzyxnew FdddTransF ),,(

Homogeneous Transformation

21

10

20 AAA

ii A1

Composite Homogeneous Transformation Matrix

0x

0z

0y

10 A

21A

1x

1z

1y 2x

2z2y

?Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices

Example 8: homogeneous transformation based on geometry

• For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5

1000zzzz

yyyy

xxxx

pasn

pasn

pasn

F

1000

010

100

0001

10

da

ceA

ii A1

iA0

0x 0y

0z

a

b

c

d

e

1x

1y

1z

2z2x

2y

3y3x

3z

4z

4y4x

5x5y

5z

1000

0100

001

010

20 ce

b

A

1000

0001

100

010

21 da

b

A

Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?

Orientation Representation

• Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body.

• Any easy way?

101333 PR

F

Euler Angles Representation

Orientation Representation• Euler Angles Representation ( , , )

– Many different types– Description of Euler angle representations

Euler Angle I Euler Angle II Roll-Pitch-Yaw

Sequence about OZ axis about OZ axis about OX axis

of about OU axis about OV axis about OY axis

Rotations about OW axis about OW axis about OZ axis

Euler Angle I, Animated

x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Orientation Representation• Euler Angle I

100

0cossin

0sincos

,

cossin0

sincos0

001

,

100

0cossin

0sincos

''

'

w

uz

R

RR

Euler Angle I

cossincossinsin

sincos

coscoscos

sinsin

cossincos

cossin

sinsincoscossin

sincos

cossinsin

coscos

''' wuz RRRR

Resultant eulerian rotation matrix:

Euler Angle II, Animated

x

y

z

u'

v'

=v"

w"

w'=

u"

v"'

u"'

w"'=

Note the opposite (clockwise) sense of the third rotation, f.

Orientation Representation

• Matrix with Euler Angle II

cossinsinsincos

sinsin

coscossin

coscos

coscossin

sincos

sincoscoscossin

cossin

coscoscos

sinsin

Quiz: How to get this matrix ?

Orientation Representation

• Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?

The City College of New York

38

The City College of New York

39

Kinematics of Robot Manipulator

Jizhong XiaoDepartment of Electrical Engineering

City College of New Yorkjxiao@ccny.cuny.edu

Introduction to ROBOTICS