Mohiqbal - 4 Kinematics

7/23/2019 Mohiqbal - 4 Kinematics

http://slidepdf.com/reader/full/mohiqbal-4-kinematics 1/44

The City College of New York

Pengantar Robotika – Universitas Gunadarma 1

Mohammad IqbalBased-on Slide Jizhong Xiao

Robot Kinematics II

Introduction to ROBOTICS





Outline

• Review

– Manipulator Specifications• Precision, Repeatability

– Homogeneous Matrix

• Denavit-Hartenberg (D-H)Representation

• Kinematics Equations• Inverse Kinematics





Review

• Manipulator, Robot arms, Industrial robot

– A chain of rigid bodies (links) connected by

joints (revolute or prismatic)

• Manipulator Specification

– DOF, Redundant Robot

– Workspace, Payload

– Precision

– Repeatability

How accurately a specified point can be reached

How accurately the same position can be reached

if the motion is repeated many times





Review• Manipulators:

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





Review•

Basic Rotation Matrix

uvw xyz RP P

xyz uvw QP P

T R RQ 1

uvw

w

v

u

z

y

x

xyz RP

p

p

p

p

p

p

P

wzvzuz

wyvyuy

wxvxux

k k jk ik

k j j ji j

k i jiii

x

z

y

v

w

P

u





Basic Rotation Matrices – Rotation about x-axis with

– Rotation about y-axis with

– Rotation about z-axis with

uvw xyz RP P

C S

S C x Rot

0

0

001

),(

0

010

0

),(

C S

S C

y Rot

100

0

0

),(

C S

S C

z Rot


http://slidepdf.com/reader/full/mohiqbal-4-kinematics 7/44The City College of New YorkPengantar Robotika – Universitas Gunadarma 7

Review•

Coordinate transformation from { B

} to { A

}

• Homogeneous transformation matrix

'o A P B

B

A P A r r Rr

1101 31

' P Bo A

B

A P A

r r Rr

1010

1333

31

' P Rr RT

o A

B

A

B

A

Position

vector

Rotation

matrix

Scaling



Review•

Homogeneous Transformation – Special cases

1. Translation

2. Rotation

10

0

31

13 B

A

B

A RT

10 31

'

33

o A

B A r I T



Review

• Composite Homogeneous TransformationMatrix

• 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



Review• Homogeneous Representation

– A point in space

– A frame in space

3 R

1000

1000 z z z z

y y y y

x x x x

pa sn

pa sn

pa sn

P a sn F

x

y

z ),,( z y x p p p P

n

sa

1

z

y

x

p

p

p

P Homogeneous coordinate of P w.r.t. OXYZ

3 R



Review

• Orientation Representation(Euler Angles)

– Description of Yaw, Pitch, Roll

•

A rotation of about the OXaxis ( ) -- yaw

• A rotation of about the OY

axis ( ) -- pitch

•

A rotation of about the OZaxis ( ) -- roll

X

Y

Z

, x R

, y R

, z R

yaw

pitch

roll



Model Kinematik•

Forward (direct) Kinematics

• Inverse Kinematics

),,( 21 nqqqq

),,,,,( z y xY

x

y

z

Direct Kinematics

Inverse Kinematics

Position and Orientation

of the end-effector

Joint

variables



Robot Link dan Joint



Denavit-Hartenberg Convention

• Number the joints from 1 to n starting with the base and ending with

the end-effector.• Establish the base coordinate system. Establish a right-handed

orthonormal coordinate system at the supporting base

with axis lying along the axis of motion of joint 1.

• Establish joint axis. Align the Zi with the axis of motion (rotary or

sliding) of joint i+1.

• Establish the origin of the ith coordinate system. Locate the origin of

the ith coordinate at the intersection of the Zi & Zi-1 or at the

intersection of common normal between the Zi & Zi-1 axes and the Zi

axis.

• Establish X i axis. Establish or along the

common normal between the Zi-1 & Zi axes when they are parallel.

• Establish Y i axis. Assign to complete the

right-handed coordinate system.

• Find the link and joint parameters

),,( 000 Z Y X

iiiii Z Z Z Z X 11 /)(

iiiii X Z X Z Y /)(

0 Z



Contoh I

• 3 Revolute Joints

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0O

Joint 1

Joint 2

Joint 3

Link 1 Link 2



Link Coordinate Frames• Assign Link Coordinate Frames:

– To describe the geometry of robot motion, we assign a Cartesian

coordinate frame (Oi, Xi,Yi,Zi) to each link, as follows:

• establish a right-handed orthonormal coordinate frame O0 at

the supporting base with Z0 lying along joint 1 motion axis.

• the Zi axis is directed along the axis of motion of joint (i + 1),that is, link (i + 1) rotates about or translates along Zi;

Link 1 Link 2

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0OJoint 1

Joint 2

Joint 3



Link Coordinate Frames – Locate the origin of the ith coordinate at the intersection

of the Zi & Zi-1 or at the intersection of common normal

between the Zi & Zi-1 axes and the Zi axis.

– the Xi axis lies along the common normal from the Zi-1

axis to the Zi axis , (if Zi-1 is

parallel to Zi , then Xi is specified arbitrarily, subject only

to Xi being perpendicular to Zi );


a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0OJoint 1

Joint 2

Joint 3



Link Coordinate Frames – Assign to complete the right-

handed coordinate system.

• The hand coordinate frame is specified by the geometry

of the end-effector. Normally, establish Zn along the

direction of Zn-1 axis and pointing away from the robot;

establish Xn such that it is normal to both Zn-1 and Zn axes. Assign Yn to complete the right-handed coordinate

system.

iiiii X Z X Z Y /)(

nO

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0OJoint 1

Joint 2

Joint 3



The City College of New YorkPengantar Robotika – Universitas Gunadarma 19

Parameter Link dan Joint• Joint angle : the angle of rotation from the X

i-1

axis to

the Xi axis about the Zi-1 axis. It is the joint variable if joint iis rotary.

• Joint distance : the distance from the origin of the (i-1)coordinate system to the intersection of the Zi-1 axis and

the Xi axis along the Zi-1 axis. It is the joint variable if joint iis prismatic.

• Link length : the distance from the intersection of the Zi-1 axis and the Xi axis to the origin of the ith coordinate

system along the Xi axis.

• Link twist angle : the angle of rotation from the Zi-1 axisto the Zi axis about the Xi axis.

i

id

ia

i




Contoh I

Joint i i a i d i i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

D-H Link Parameter Table

: rotation angle from Xi-1

to Xi about Z

i-1

i

: distance from origin of (i-1) coordinate to intersection of Zi-1

& Xi along Z

i-1

: distance from intersection of Zi-1 & Xi to origin of i coordinate along X

i

id

: rotation angle from Zi-1

to Zi about X

i

ia

i

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0OJoint 1

Joint 2

Joint 3




Contoh II: PUMA 260


iiiii X Z X Z Y /)(

1 2

3

4

5 6

0 Z

1 Z

2 Z

3 Z

4 Z 5 Z

1O

2O

3O

5O4O

6O

1 X 1Y

2 X 2Y

3 X

3Y

4 X

4Y

5 X 5Y

6 X

6Y

6 Z

1. Number the joints

2. Establish base frame

3. Establish joint axis Zi

4. Locate origin, (intersect.

of Zi & Zi-1) OR (intersect

of common normal & Zi )

5. Establish Xi,Yi

PUMA 260

t




Link Parameters

1 2

3

4

5 6

0 Z

1 Z

2 Z

3 Z

4 Z 5 Z

1O

2O

3O

5O4O

6O

1 X 1Y

2 X 2Y

3 X

3Y

4 X

4Y

5 X 5Y

6 X

6Y

6 Z

: angle from Zi-1

to Zi

about Xi

: distance from intersectionof Z

i-1& X

ito Oi along X

i

Joint distance : distance from Oi-1 to intersection of Zi-1

& Xi along Z

i-1

: angle from Xi-1

to Xi

about Zi-1

i

i

ia

id

t006

0090580-904

0

0903

802

130-901

J i

1

4

2

3

6

5

i ia id

-




Transformasi antara i-1 dan i

•

Four successive elementary transformationsare required to relate the i -th coordinate frame

to the (i -1)-th coordinate frame:

– Rotate about the Z i-1 axis an angle of i to align the

X i-1 axis with the X i axis. – Translate along the Z i-1 axis a distance of di, to bring

Xi-1 and Xi axes into coincidence.

– Translate along the Xi axis a distance of ai to bring

the two origins Oi-1 and Oi as well as the X axis intocoincidence.

– Rotate about the Xi axis an angle of αi ( in the right-

handed sense), to bring the two coordinates into

coincidence.




Transformasi antara i-1 dan i

• D-H transformation matrix for adjacent coordinate

frames, i and i-1.

– The position and orientation of the i -th frame coordinate

can be expressed in the (i-1)th frame by the following

homogeneous transformation matrix:

1000

0

),(),(),(),( 111

iii

iiiiiii

iiiiiii

iiiiiiiiii

d C S

S aC S C C S

C aS S S C C

x Ra xT z Rd z T T

Source coordinate

ReferenceCoordinate




Persamaan Kinematik• Forward Kinematics

– Given joint variables

– End-effector position & orientation

• Homogeneous matrix

– specifies the location of the ith coordinate frame w.r.t.the base coordinate system

– chain product of successive coordinate transformation

matrices of

100010

000

1

2

1

1

00

nnn

n

n

n

P a sn P R

T T T T

),,(21 n

qqqq

),,,,,( z y xY

iiT 1

nT 0

Orientation

matrix

Position

vector




Persamaan Kinematik

• Other representations – reference from, tool frame

– Yaw-Pitch-Roll representation for orientation

tool

n

n

ref

tool

ref H T BT 0

0

,,, x y z R R RT

1000

0100

00

00

C S

S C

1000

00

0010

00

C S

S C

1000

00

00

0001

C S

S C




Representasi forward kinematics

• Forward kinematics

z

y

x

p p

p

6

5

4

3

2

1

1000

z z z z

y y y y

x x x x

pa sn

pa sn pa sn

T

• Transformation Matrix




Representasi forward kinematics

•

Yaw-Pitch-Roll representation for orientation

1000

0

z

y

x

n

pC C S C S

pS C C S S C C S S S C S

pS S C S C C S S S C C C

T

1000

0

z z z z

y y y y

x x x x

n

pa sn

pa sn

pa sn

T

)(sin 1

z n

)

cos

(cos 1

z a

)cos

(cos 1

xn

Problem? Solution is inconsistent and ill-conditioned!!




Representasi Yaw-Pitch-Roll

,,, x y z

R R RT

1000

0100

00

00

C S

S C

1000

00

0010

00

C S

S C

1000

00

00

0001

C S

S C

1000

0

0

0

z z z

y y y

x x x

a sn

a sn

a sn





,,

1

, x y z R RT R

1000

0100

00

00

C S

S C

1000

00

0010

00

C S

S C

1000

00

00

0001

C S

S C

1000

0

0

0

z z z

y y y

x x x

a sn

a sn

a sn

(Equation A)





0cossin y x nn

sincossincos

z

y x

nnn

sincossin

coscossin

y x

y x

aa

s s

• Compare LHS and RHS of Equation A, we have:

),(2tan x y nna

)sincos,(2tan y z z nnna

)cossin,cos(sin2tan y x y x s saaa




Model Kinematic

• Steps to derive kinematics model:

– Assign D-H coordinates frames

– Find link parameters

– Transformation matrices of adjacent joints

– Calculate Kinematics Matrix

– When necessary, Euler angle representation




Contoh

Joint i i a i d i i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X3 3O

2O1O0OJoint 1

Joint 2

Joint 3




Contoh Joint i i a i d i i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

1000

0100

0cosθsinθ

0sinθcosθ

00

00

0

00

00

1sin

cos

a

a

T

1000

000

sinθcosθ

1

1

11

sincos0cossin0

111

111

2

aa

T

))()(( 210

3213

0 T T T T

1000

0sinθcosθ 22

2

2

223

100

00cossin

0

d T

1000

01

iii

iiiiiii

iiiiiii

ii

d C S

S aC S C C S

C aS S S C C

T




Contoh: Puma 560




Contoh : Puma 560




Parameter Koordinat Link

Joint i i i a i (mm) d i (mm

1 1 -90 0 0

2 2 0 431.8 149.09

3 3 90 -20.32 0

4 4 -90 0 433.07

5 5 90 0 0

6 6 0 0 56.25

PUMA 560 robot arm link coordinate parameters




Contoh : Puma 560




Inverse Kinematics• Given a desired position (P)

& orientation (R) of the end-

effector

• Find the joint variables

which can bring the robot

the desired configuration

),,( 21 nqqqq

x

y

z




Inverse Kinematics

• More difficult – Systematic closed-form

solution in general is not

available

– Solution not unique

• Redundant robot

• Elbow-up/elbow-down

configuration – Robot dependent

(x , y)




Inverse Kinematics

6

5

4

3

2

1

6

5

5

4

4

3

3

2

2

1

1

0

1000

T T T T T T pa sn

pa sn

pa sn

T z z z z

y y y y

x x x x

• Transformation Matrix

Special cases make the closed-form arm solution possible:

1. Three adjacent joint axes intersecting (PUMA, Stanford)

2. Three adjacent joint axes parallel to one another (MINIMOVER)



Terima Kasih

x

y z

x

y z

x

y z

x

z

y

Materi Selanjutnya : Mobot

Mohiqbal - 4 Kinematics

Documents

Transcript of Mohiqbal - 4 Kinematics