introduction to robotics - gunadarmamohiqbal.staff.gunadarma.ac.id/downloads/files/82987/... ·...

The City College of New York

Pengantar Robotika – Universitas Gunadarma 1

Mohammad Iqbal Based-on slide Jizhong Xiao

Kinematics of Robot Manipulator

Introduction to ROBOTICS



Outline • Review

• Robot Manipulators – Robot Configuration

– Robot Specification • Number of Axes, DOF

• Precision, Repeatability

• Kinematics – Preliminary

• World frame, joint frame, end-effector frame

• Rotation Matrix, composite rotation matrix

• Homogeneous Matrix

– Direct kinematics • Denavit-Hartenberg Representation

• Examples

– Inverse kinematics



Review • What is a robot?

– By general agreement a robot is: • A programmable machine that imitates the actions or

appearance of an intelligent creature–usually a human.

– To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings

2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects

3) Re-programmable: can do different things

4) Function autonomously and/or interact with human beings

• Why use robots?

4A: Automation, Augmentation, Assistance, Autonomous

4D: Dangerous, Dirty, Dull, Difficult

–Perform 4A tasks in 4D environments



Manipulators

• Robot arms, industrial robot

– Rigid bodies (links) connected

by joints

– Joints: revolute or prismatic

– Drive: electric or hydraulic

– End-effector (tool) mounted

on a flange or plate secured

to the wrist joint of robot



Manipulators • Robot Configuration:

Cartesian: PPP Cylindrical: RPP Spherical: RRP

SCARA: RRP

(Selective Compliance

Assembly Robot Arm)

Articulated: RRR

Hand coordinate:

n: normal vector; s: sliding vector;

a: approach vector, normal to the

tool mounting plate



Manipulators

• Motion Control Methods

– Point to point control

• a sequence of discrete points

• spot welding, pick-and-place, loading & unloading

– Continuous path control

• follow a prescribed path, controlled-path motion

• Spray painting, Arc welding, Gluing



Manipulators

• Robot Specifications

– Number of Axes

• Major axes, (1-3) => Position the wrist

• Minor axes, (4-6) => Orient the tool

• Redundant, (7-n) => reaching around

obstacles, avoiding undesirable

configuration

– Degree of Freedom (DOF)

– Workspace

– Payload (load capacity)

– Precision v.s. Repeatability Which one is more important?



What is Kinematics

• Forward kinematics

Given joint variables

End-effector position and orientation, -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyxY x

y

z



Apa itu Kinematics

• Inverse kinematics

End effector position

and orientation

Joint variables -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyx

x

y

z



Contoh 1

0x

0y

1x1y

)/(cos

kinematics Inverse

sin

cos

kinematics Forward

0

1

0

0

lx

ly

lx

l



Pendahuluan

• Robot Reference Frames

– World frame

– Joint frame

– Tool frame

x

yz

x

z

y

W R

P T



Pendahuluan

• Coordinate Transformation

– Reference coordinate frame

OXYZ

– Body-attached frame O’uvw

wvu kji wvuuvw pppP

zyx kji zyxxyz pppP

x

y

z

P

u

v

w

O, O’

Point represented in OXYZ:

zwyvxu pppppp

T

zyxxyz pppP ],,[

Point represented in O’uvw:

Two frames coincide ==>



Pendahuluan

• Mutually perpendicular

• Unit vectors

Properties of orthonormal coordinate frame

0

0

0

jk

ki

ji

1||

1||

1||

k

j

i

Properties: Dot Product

Let and be arbitrary vectors in and be

the angle from to , then

3R

cosyxyx

x y

x y



Pendahuluan

• 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?



Pendahuluan

• 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



Pendahuluan • 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

w

P

u

CS

SCxRot

0

0

001

),(



Pendahuluan

• Is it True?

– 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

w

P

u



Dasar Matrik Rotasi – 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



Pendahuluan

• Basic Rotation Matrix

– Obtain the coordinate of from the coordinate

of

uvwxyz RPP

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

R

xyzuvw QPP

TRRQ 1

3

1 IRRRRQR T

uvwP

xyzP



Contoh 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.

)2,3,4(uvwa

2

964.4

598.0

2

3

4

100

05.0866.0

0866.05.0

)60,( uvwxyz azRota



Contoh 3

• 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



Matrik Rotasi Komposit

• 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



Contoh 4

• 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



Koordinat Transformasi

• 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}



Koordinat Transformasi • Two Special Cases

1. Translation only

– Axes of {B} and {A} are

parallel

2. Rotation only

– Origins of {B} and {A}

are coincident

1BAR

'oAPB

B

APA rrRr

0' oAr



Representasi Homogen • Coordinate transformation from {B} to {A}

• Homogeneous transformation matrix

'oAPB

B

APA rrRr

1101 31

' PBoA

B

APA rrRr

1010

1333

31

' PRrRT

oA

B

A

B

A

Position

vector

Rotation

matrix

Scaling



Transformasi Homogen • Special cases

1. Translation

2. Rotation

10

0

31

13B

A

B

A RT

10 31

'

33

oA

B

A rIT



Contoh 5 • 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

v

w

O, O’ h x

y

z

P

u

v

w

O, O’



Contoh 6 • Rotation about the X-axis by

1000

00

00

0001

),(

CS

SCxRot

x

z

y

v

w

P

u

11000

00

00

0001

1

w

v

u

p

p

p

CS

SC

z

y

x



Transformasi Homogen

• 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



Contoh 7

• Find the homogeneous transformation matrix

(T) for the following operations:

:

axis OZabout ofRotation

axis OZ along d ofn Translatio

axis OX along a ofn Translatio

axis OXabout Rotation

Answer

44,,,, ITTTTT xaxdzz

1000

00

00

0001

1000

0100

0010

001

1000

100

0010

0001

1000

0100

00

00

CS

SC

a

d

CS

SC



Representasi Homogen

• A frame in space (Geometric

Interpretation)

x

y

z

),,( zyx pppP

1000

zzzz

yyyy

xxxx

pasn

pasn

pasn

F

n

sa

10

1333 PRF

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

(X’)

(y’)

(z’)




• 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 ),,(




2

1

1

0

2

0 AAA

Composite Homogeneous Transformation Matrix

0x

0z

0y

1

0 A2

1A

1x

1z

1y2x

2z2y

?

i

i A1 Transformation matrix for adjacent coordinate frames

Chain product of successive

coordinate transformation matrices



Contoh 8 • For the figure shown below, find the 4x4 homogeneous transformation

matrices and for i=1, 2, 3, 4, 5

1000

zzzz

yyyy

xxxx

pasn

pasn

pasn

F

i

i A1 iA0

0x0y

0z

a

b

c

d

e

1x

1y

1z

2z

2x

2y

3y3x

3z

4z

4y4x

5x

5y

5z

1000

010

100

0001

1

0

da

ceA

1000

0100

001

010

2

0ce

b

A

1000

0001

100

010

2

1da

b

A

Can you find the answer by observation

based on the geometric interpretation of

homogeneous transformation matrix?



Representasi Orientasi

• Rotation matrix representation needs 9

elements to completely describe the

orientation of a rotating rigid body.

• Any easy way?

10

1333 PRF

Euler Angles 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



x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Ilustrasi Sudut Euler I




• Euler Angle I

100

0cossin

0sincos

,

cossin0

sincos0

001

,

100

0cossin

0sincos

''

'

w

uz

R

RR



Sudut Euler I

cossincossinsin

sincos

coscoscos

sinsin

cossincos

cossin

sinsincoscossin

sincos

cossinsin

coscos

''' wuz RRRR

Resultant eulerian rotation matrix:



Ilustrasi Sudut Euler II

x

y

z

u'

v'

=v"

w"

w'=

u"

v"'

u"'

w"'=

Note the opposite

(clockwise) sense of the

third rotation, .




• Matrix with Euler Angle II

cossinsinsincos

sinsin

coscossin

coscos

coscossin

sincos

sincoscoscossin

cossin

coscoscos

sinsin

Quiz: How to get this matrix ?




• Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?



Terima Kasih

x

yz

x

yz

x

yz

x

z

y

Bahasan Selanjutnya : kinematics II

introduction to robotics - gunadarmamohiqbal.staff.gunadarma.ac.id/downloads/files/82987/... ·...

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