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The City College of New York 1

Jizhong Xiao

Department of Electrical Engineering

City College of New [email protected]

Robot Kinematics II

Introduction to ROBOTICS

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Outline

Review

Manipulator Specifications Precision, Repeatability

Homogeneous Matrix

Denavit-Hartenberg (D-H)Representation

Kinematics Equations Inverse Kinematics

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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

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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

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Review

Basic Rotation Matrix

uvwxyz RPP

xyzuvw QPP

TRRQ 1

uvw

w

v

u

z

y

x

xyz RP

p

p

p

p

p

p

P

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

x

z

y

v

w

P

u

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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

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Review

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

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Review

Homogeneous Transformation Special cases

1. Translation

2. Rotation

10

0

31

13B

A

B

A RT

10 31

'

33

oA

BA rIT

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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

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Review Homogeneous Representation

A point in space

A frame in space

3R

1000

1000 zzzz

yyyy

xxxx

pasn

pasn

pasn

PasnF

x

y

z ),,( zyx pppP

n

sa

1

z

y

x

p

p

p

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

3R

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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

,xR

,yR

,zR

yaw

pitch

roll

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Quiz 1 How to get the resultant rotation matrix for YPR?

X

Y

Z

,,, xyz RRRT

1000

0100

00

00

CS

SC

1000

00

0010

00

CS

SC

1000

00

00

0001

CS

SC

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Quiz 2

Geometric Interpretation?

10

1333PR

T

10

1 PRRTTT

44

1

10

0

1010

I

RRPRPRRTT

TTT

Position of the origin of OUVW

coordinate frame w.r.t. OXYZ frame

Inverse of the rotation submatrix

is equivalent to its transpose

Position of the origin of OXYZ

reference frame w.r.t. OUVW frame

Orientation of OUVW coordinate

frame w.r.t. OXYZ frame

Inverse Homogeneous Matrix?

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Kinematics Model

Forward (direct) Kinematics

Inverse Kinematics

),,( 21 nqqqq

),,,,,( zyxY

x

y

z

Direct Kinematics

Inverse Kinematics

Position and Orientation

of the end-effector

Joint

variables

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Robot Links and Joints

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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 Xiaxis. Establish or along the

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

Establish Yiaxis. Assign to complete the

right-handed coordinate system.

Find the link and joint parameters

),,( 000 ZYX

iiiii ZZZZX 11 /)(

iiiii XZXZY /)(

0Z

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Example I

3 Revolute Joints

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O

1O

0O

Joint 1

Joint 2

Joint 3

Link 1 Link 2

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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 O0at

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

the Ziaxis 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

X33

O

2O

1O

0O

Joint 1

Joint 2

Joint 3

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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 Xiaxis lies along the common normal from the Zi-1

axis to the Ziaxis , (if Zi-1 is

parallel to Zi, then Xiis specified arbitrarily, subject only

to Xibeing perpendicular to Zi);

iiiii ZZZZX 11 /)(

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0OJoint 1

Joint 2

Joint 3

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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 Znaxes. Assign Yn to complete the right-handed coordinate

system.

iiiii XZXZY /)(

nO

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33

O

2O

1O

0O

Joint 1

Joint 2

Joint 3

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Link and Joint Parameters Joint angle : the angle of rotation from the X

i-1

axis to

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

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

system along the Xiaxis.

Link twist angle : the angle of rotation from the Zi-1 axisto the Ziaxis about the Xiaxis.

i

id

ia

i

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Example I

Joint i i ai di i1 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 Zi-1

i

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

& Xialong Z

i-1

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

i

id

: rotation angle from Zi-1

to Ziabout X

i

ia

i

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33

O

2O1O0OJoint 1

Joint 2

Joint 3

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Example II: PUMA 260

iiiii ZZZZX 11 /)(

iiiii XZXZY /)(

1

2

3

4

5

6

0Z

1Z

2Z

3Z

4Z5Z

1O

2O

3O

5O4O

6O

1X1Y

2X2

Y

3X

3Y

4X

4Y

5X

5Y

6X

6Y

6Z

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

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Link Parameters

1

2

3

4

5

6

0Z

1Z

2Z

3Z

4Z5Z

1O

2O

3O

5O4O

6O

1X1Y

2X2

Y

3X

3Y

4X

4Y

5X

5Y

6X

6Y

6Z

: angle from Zi-1

to Zi

about Xi

: distance from intersectionof Z

i-1& X

ito Oialong X

i

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

& Xialong Z

i-1

: angle from Xi-1

to Xi

about Zi-1

i

i

ia

i

d

t006

0090580-904

0

0903

802

130-901

J i

1

4

2

3

6

5

i ia id

-l

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Transformation between i-1 and 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 ofi to align the

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

Xi-1 and Xiaxes into coincidence.

Translate along the Xiaxis a distance ofaito bring

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

Rotate about the Xiaxis an angle ofi( in the right-

handed sense), to bring the two coordinates into

coincidence.

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Transformation between i-1 and i

D-H transformation matrix for adjacent coordinate

frames, iand 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

iiiiiiii

i

i

dCS

SaCSCCS

CaSSSCC

xRaxTzRdzTT

Source coordinate

ReferenceCoordinate

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Kinematic Equations 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

PasnPR

TTTT

),,(21 n

qqqq

),,,,,( zyxY

i

iT 1

nT0

Orientation

matrix

Position

vector

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Kinematics Equations

Other representations reference from, tool frame

Yaw-Pitch-Roll representation for orientation

tool

n

n

ref

tool

ref HTBT 00

,,, xyz RRRT

1000

0100

00

00

CS

SC

1000

00

0010

00

CS

SC

1000

00

00

0001

CS

SC

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Representing forward kinematics

Forward kinematics

z

y

x

pp

p

6

5

4

3

2

1

1000

zzzz

yyyy

xxxx

pasn

pasnpasn

T

Transformation Matrix

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Representing forward kinematics

Yaw-Pitch-Roll representation for orientation

1000

0

z

y

x

n

pCCSCS

pSCCSSCCSSSCS

pSSCSCCSSSCCC

T

1000

0

zzzz

yyyy

xxxx

n

pasn

pasn

pasn

T

)(sin 1 zn

)

cos

(cos 1

za

)cos

(cos 1

x

n

Problem? Solution is inconsistent and ill-conditioned!!

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atan2(y,x)

x

y

yandxfor

yandxfor

yandxfor

yandxfor

xya

090

90180

18090

900

),(2tan

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Yaw-Pitch-Roll Representation

,,, xyz RRRT

1000

0100

00

00

CS

SC

1000

00

0010

00

CS

SC

1000

00

00

0001

CS

SC

1000

0

0

0

zzz

yyy

xxx

asn

asn

asn

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,,

1

, xyz RRTR

1000

0100

00

00

CS

SC

1000

00

0010

00

CS

SC

1000

00

00

0001

CS

SC

1000

0

0

0

zzz

yyy

xxx

asn

asn

asn

(Equation A)

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0cossin yx nn

sincossincos

z

yx

nnn

sincossin

coscossin

yx

yx

aa

ss

Compare LHS and RHS of Equation A, we have:

),(2tan xy nna

)sincos,(2tan yzz nnna

)cossin,cos(sin2tan yxyx ssaaa

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Kinematic Model

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

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Example

Joint i i ai di i1 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

X33O

2O

1O

0O

Joint 1

Joint 2

Joint 3

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ExampleJoint i i ai di i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

1000

0100

0cossin

0sincos

00

00

0

00

00

1sin

cos

a

a

T

1000

000

sincos

1

1

11

sincos0cossin0

111

111

2

aa

T

))()(( 2103213

0 TTTT

1000

0sincos 22

2

2

223

100

00cossin

0

dT

1000

01

iii

iiiiiii

iiiiiii

i

idCS

SaCSCCS

CaSSSCC

T

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Example: Puma 560

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Example: Puma 560

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Link Coordinate Parameters

Joint i i i ai(mm) di(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

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Example: Puma 560

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Example: Puma 560

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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

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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)

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Inverse Kinematics

6

5

4

3

2

1

6

5

5

4

4

3

3

2

2

1

1

0

1000

TTTTTTpasn

pasn

pasn

Tzzzz

yyyy

xxxx

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)

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Thank you!

x

yz

x

yz

x

yz

x

z

y

Homework 2 posted on the web.

Due: Sept. 23, 2008

Next class: Inverse Kinematics, Jocobian

Matrix, Trajectory planning

puma 1 et 2

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