kinematik robot

1Tugino, ST [email protected]

Kinematika Manipulator Robot

1

Jurusan Teknik Elektro STTNASYogyakarta

Pendahuluan Robot Manipulators

Konfigurasi Robot S ifik i R b t Spesifikasi Robot Jumlah Axis, DOF(Degree Of Freedom) Precision, Repeatability

Kinematics Preliminary

World frame, joint frame, end-effector frame Rotation Matrix, composite rotation matrix Homogeneous Matrix

Tugino, ST MT Sekolah Tinggi Teknologi Nasional Yogyakarta 2

Homogeneous Matrix Direct kinematics

Denavit-Hartenberg Representation Examples

Inverse kinematics

2Manipulators

Lengan Robot , Robot industri Rigid bodies (links) connected 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


j

Manipulators R(revolute) or P( prismatic)

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

3Robots Cartesian

Cartesian robot Arm moves in 3 linear Arm moves in 3 linear

axes. X,y,z axes

z


xy


4Manipulators

Motion Control MethodsP i t t i t t l 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


Spray painting, Arc welding, Gluing

Manipulators

Robot Specifications Number of Axes 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?

5Kinematik

0ykinematics Forward

x

0y

1x1y

kinematics Inverse

sincos

0

0

lylx

==

l


0x)/(cos 01 lx=


6Tugino, ST MT Sekolah Tinggi Teknologi Nasional Yogyakarta 11

A Simple Example

Revolute and Prismatic Joints Finding : y

Y

S

Combined

(x , y)

)xyarctan( =

More Specifically:

)xy(2arctan = arctan2() specifies that its in the first quadrant


1X

SFinding S:

)y(xS 22+=

72

(x , y)

Inverse Kinematics of a Two Link Manipulator

Given: l1, l2 , x , y

Find: 1, 2

1

l2

l1

Redundancy:A unique solution to this problem

does not exist. Notice, that using the givens two solutions are possible. Sometimes no solution is possible.


(x , y)

Preliminary

Robot Reference FramesW ld f World frame

Joint frame Tool frame

x

yz

z

y T


x

y

W R

PT

8Preliminary Coordinate Transformation

Reference coordinate frame OXYZ zOXYZ

Body-attached frame Ouvw

zyx kji zyxxyz pppP ++=r

y

zP

vw

Point represented in OXYZ:T

zyxxyz pppP ],,[=


wvu kji wvuuvw pppP ++=r

zyx j zyxxyz pppx

uO, O

zwyvxu pppppp ===

Point represented in Ouvw:

Two frames coincide ==>

PreliminaryProperties: Dot Product Let and be arbitrary vectors in and be 3R x y

Mutually perpendicular Unit vectorsProperties of orthonormal coordinate frame

0= ji vv 1|| v

the angle from to , thencosyxyx =

x y


00

0

==

jkkiji

vvvv

1||1||1||

===

kji

vv

9Preliminary

Coordinate Transformation Rotation only z Rotation only

wvu kji wvuuvw pppP ++=r y

zP

zyx kji zyxxyz pppP ++=r

u

vw


xuvwxyz RPP = u

How to relate the coordinate in these two frames?

PreliminaryBasic Rotation , , and represent the projections of xp Pyp zp

onto OX, OY, OZ axes, respectively

Since

x y z

wvux pppPp wxvxuxx kijiiii ++==wvu kji wvu pppP ++=


wvuy pppPp wyvyuyy kjjjijj ++==wvuz pppPp wzvzuzz kkjkikk ++==

10

wvux pppPp wxvxuxx kijiiii ++==wvuy pppPp wyvyuyy kjjjijj ++==

wvuz pppPp wzvzuzz kkjkikk ++==

uxpp wxvxux kijiii


=

w

v

z

y

pp

pp

wzvzuz

wyvyuy

kkjkikkjjjij

PreliminaryBasic Rotation Matrix

uxpp wxvxux kijiii

Rotation about x-axis with

=

w

v

z

y

pp

pp

wzvzuz

wyvyuy

kkjkikkjjjij

zw

Tugino, ST MT Sekolah Tinggi Teknologi Nasional Yogyakarta 20x

y

vP

u

=

CSSCxRot

00

001),(

11

Preliminary

Is it True? Rotation about x axis with

cossin0sincos0001

w

v

u

z

y

x

ppp

ppp

=

z

v

wP


cossin

sincos

wvz

wvy

ux

pppppp

pp

+==

=

x

yu

Basic Rotation Matrices Rotation about x-axis with

= SCxRot 0001

),(

Rotation about y-axis with

R t ti b t i ith

CS0

0

0100

),(

=

CS

SCyRot


Rotation about z-axis with

uvwxyz RPP =

=

10000

),(

CSSC

zRot

12

Preliminary Basic Rotation Matrix

wxvxux kijiii

Obtain the coordinate of from the coordinate of

uvwxyz RPP =

=wzvzuz

wyvyuy

kkjkikkjjjijR

QPP =

uvwP

xyzP

xu

pp zuyuxu kijiii

Dot products are commutative!


xyzuvw QPP =TRRQ == 1

31 IRRRRQR T ===

13

Example 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

)2,3,4(=xyza

uvwapoint w.r.t. the rotated OU V W coordinate system if it has been rotated 60 degree about OZ axis.

uvwa

=

598440866050

)60,( xyzT

uvw azRota


=

=

2964.1

598.4

234

10005.0866.00866.05.0

Composite Rotation Matrix A sequence of finite rotations matrix multiplications do not commutep 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


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

14

Example 4 Find the rotation matrix for the following

operations:

axis OUabout Rotation axisOW about Rotation

axis OYabout Rotation

+

=

=

=

SCCCSCSSSCCSCSSCC

CSSCCS

SCuRotwRotIyRotR

00

001

10000

C0S-010

S0C),(),(),( 3

Tugino, ST MT Sekolah Tinggi Teknologi Nasional Yogyakarta 27Post-multiply if rotate about the OUVW axes Pre-multiply if rotate about the OXYZ axes

...Answer + SSSCCSCCSSCS

Example 5 Translation along Z-axis with h:

00100001

00100001 ppx uu

=

1000100

0010),(

hhzTrans

+=

=

111000100

0010

1hp

ppp

hzy

w

v

w

v

y

z P

w

zP


x

y

u

vw

O, Ohx

y

u

vw

O, O

15

Example 6 Rotation about the X-axis by

0001

0001 upx

=10000000

),(

CSSC

xRot

z

v

wP

=

110000000

1w

v

pp

CSSC

zy

Tugino, ST MT Sekolah Tinggi Teknologi Nasional Yogyakarta 29x

y

v

u

Homogeneous Transformation

Composite Homogeneous Transformation MatrixMatrix

Rules: Transformation (rotation/translation) w.r.t

(X,Y,Z) (OLD FRAME), using pre-multiplication

Transformation (rotation/translation) w r t


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

16

Example 7 Find the homogeneous transformation matrix (T) for

the following operation:

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

axis OXabout Rotation

ITTTTT


:Answer44,,,, = ITTTTT xaxdzz

=

100000000001

100001000010

001

1000100

00100001

100001000000

CSSC

a

dCSSC

Homogeneous Representation

A frame in space (Geometric Interpretation)

z),,( zyx pppP

y

z

yyyy

xxxx

pasnpasn

F

ns

a

= 10

1333 PRF(X)

(y)(z)


x

=

1000zzzz

yyyy

pasnF

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

17

Homogeneous Transformation

Translation s

a

y

z

ns

a ns

+

=

10001000100010001

xxxxx

zzzz

yyyy

xxxx

z

y

x

new

dpasn

pasnpasnpasn

ddd

F


++=1000

zzzzz

yyyyy

dpasndpasn

oldzyxnew FdddTransF = ),,(

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


of about OU axis about OV axis about OY axis

Rotations about OW axis about OW axis about OZ axis

18

Orientation Representation Euler Angle I

0010sincos

=

=

=

0cossin0sincos

,cossin0sincos0001

,1000cossin0sincos

'

uz

R

RR


=1000cossin'' wR

Euler Angle IResultant eulerian rotation matrix:

=

sincossinsincossin

sinsincoscossin

sincoscossinsin

coscos''' wuz RRRR


+

+

cossincossinsin

sincoscoscoscos

sinsincossincos

cossin

19

Euler Angle II, Animated

zw'=

yv'

=v"

w"

v"'

"'

w"'=


xu'

u"

u"'

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

zw'=

Euler Angle I, Animated

yv'

v "

w"v'"

w'"=


xu'

=u"

u'"

20

Orientation Representation Matrix with Euler Angle II

iii

+

+

sinsincoscossin

coscoscoscossin

sincos

sincoscoscossin

cossincoscoscos

sinsin


cossinsinsincosQuiz: How to get this matrix ?

Orientation Representation Description of Roll Pitch Yaw

Z

Y


X

Quiz: How to get rotation matrix ?

21

Terimakasih

x

yz

x

yz

x

yzz

y


x

Coordinate Transformations position vector of P in {B} is transformed to position vector of Pto 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}

22

Coordinate Transformations Two Special Cases

1 T l ti l

'oAPBB

APA rrRr +=1. Translation only Axes of {B} and {A} are

parallel

2. Rotation only

1=BAR


y Origins of {B} and {A}

are coincident

0' =oAr

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

'oAPBB

APA rrRr +=

Homogeneous transformation matrix

=

1101 31

' PBoAB

APA rrRr


g

=

= 1010

1333

31

' PRrRT

oAB

A

BA

Position vector

Rotation matrix

Scaling

23

Homogeneous Transformation Special cases

1. Translation

2. Rotation

=

10 31

'33

oA

BA rIT


=

10

0

31

13BA

BA RT

The Geometric Solution

l

l22(x , y) Using the Law of Cosines:

+=

+=++=

2222

212

22

122

222

)cos()180cos()180cos(2)(

cos2

ll

llllyx

Cabbac

22

2

l1

1

+=

+=

21

22

21

22

21

21

2arccos

2)cos(

llllyx

llllyx

2

2

Using the Law of Cosines:

= sinsinc

Cb

B Redundant since 2 could be in the first or fourth quadrant.


=+=

+=+=

xy2arctan

yx)sin(

yx)sin(180sin

11

222

222

2

1

l

cb

+

+= xy2arctan

yx)sin(arcsin

2222

1l

Redundancy caused since 2 has two possible values

24

The Algebraic Solution

l22(x , y)

1221

11

ccx(1))cos( c

cos c

++=

=+

ll

( ) ( )+++++= =+=+ ++++ 21121221222121211212212221212222

)(sins2)(sins)(cc2)(cc

yx)2((1)

llllllll

l1

121

21211

21211

(3)sinsy(2)ccx(1)

+=+=+=

+

+llll


( ) ( )( )

+=

++=+++= ++

++++

21

22

21

22

2

2212

22

1

211211212

22

1

21121212112112121211

2yxarccos

c2

)(sins)(cc2

)()()()(

llll

llll

llllOnly Unknown

))(sin(cos))(sin(cos)sin(

))(sin(sin))(cos(cos)cos(:

abbababababa

Note

++

++==

))(sin(cos))(sin(cos)sin(

))(sin(sin))(cos(cos)cos(:

abbababababa

Note

++

++==

sinsy

)()c(c ccc

ccx

21211

2212211

21221211

21211

ll

slsllsslll

ll

+=

+=+=

+=

+

+

We know what 2 is from the previous

)c(s)s(c cscss

2211221

12221211

llllll++=

++= slide. We need to solve for 1 . Now we have two equations and two unknowns (sin 1 and cos 1 )

221122221

221

221

2211

)c(s)s()c()(xy

)c()(xc

++++=

++=

lllll

slsll

sls

Substituting for c1 and simplifying many times


( )

2222221

1

2212

22

1122221

yxx)c(ys

)c2(sx)c(

1

++=

++++=

slll

llllslll

Notice this is the law of cosines and can be replaced by x2+ y2

++= 22 222211 yx

x)c(yarcsin slll

25

Homogeneous TransformationComposite Homogeneous Transformation Matrix

1z2z 2y

0x

0z

0y1

0 A2

1 A

1x

1y 2x

?


21

10

20 AAA =

?i

i A1 Transformation matrix for adjacent coordinate framesChain product of successive coordinate transformation matrices

Example 8 For the figure shown below, find the 4x4 homogeneous transformation matrices

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

xxxxpasn

ii A1 iA0

c 0001

=

1000zzzz

yyyy

xxxx

pasnpasnp

F

a

b

c

d

e

2z

3y 3x3z

4z

4y4x

5x5y

5z

+=1000

010100

10

dace

A

=0001

100010

21 da

b

A


0x 0y

0z1x

1y

1z2x

2y

+

=10000100

001010

20 ce

b

A

1000

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

26

Orientation Representation

= 10

1333 PRF

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

Any easy way?

10


Euler Angles Representation

kinematik robot

Documents

wrist joint of robot

t t i t t

linear axes

frame ouvw z

prismatic joints

joints joints

normal vector s

wrist minor axes