city college of new york 1 jizhong xiao department of electrical engineering city college of new...

1City College of New York

Jizhong Xiao

Department of Electrical Engineering

City College of New York

[email protected]

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

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

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


Review• 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


Review• Homogeneous Transformation

– Special cases

1. Translation

2. Rotation

10

0

31

13BA

BA RT

10 31

'33

oA

BA rIT


Review

• 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


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

1z

y

x

p

p

p

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

3R


Review

• Orientation Representation (Euler Angles) – Description of Roll-Pitch-Yaw

• A rotation of about the OX axis ( ) -- yaw

• A rotation of about the OY axis ( ) -- pitch

• A rotation of about the OZ axis ( ) -- roll

X

Y

Z

,xR

,yR

,zR


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


Quiz 2• Geometric Interpretation?

•

101333 PR

T

101 PRR

TTT

441

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?


Kinematics Model• Forward (direct) Kinematics

• Inverse Kinematics

),,( 21 nqqqq

),,,,,( zyxY

x

y

z

Direct K inem atics

Inverse K inem atics

Position and O rientationof the end-effector

Jointvariables


Robot Links and Joints


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 Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.

• Establish Yi axis. Assign to complete the right-handed coordinate system.

• Find the link and joint parameters

),,( 000 ZYX

iiiii ZZZZX 11 /)(

iiiii XZXZY /)(

0Z


Example• 3 Revolute Joints

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

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

X33O

2O1O0O

Z2

Joint 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);

iiiii ZZZZX 11 /)(

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

Joint 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 XZXZY /)(

nO

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

Joint 1

Joint 2

Joint 3


Link and Joint Parameters• Joint angle : the angle of rotation from the Xi-1 axis to

the Xi axis about the Zi-1 axis. It is the joint variable if joint i is 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 i is 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 axis to the Zi axis about the Xi axis.

i

id

ia

i


Example

Joint i i ai di

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

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

id

: rotation angle from Zi-1 to Zi about Xi

iai

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

Joint 1

Joint 2

Joint 3


Transformation between i-1 and i• Four successive elementary transformations are

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 into coincidence.

– Rotate about the Xi axis an angle of αi ( in the right-handed sense), to bring the two coordinates into coincidence.


Transformation between i-1 and 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

iiiiiiiii

i

dCS

SaCSCCS

CaSSSCC

xRaxTzRdzTT

Source coordinate

ReferenceCoordinate


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

100010000

12

11

00

nnn

nn

n

PasnPR

TTTT

),,( 21 nqqqq

),,,,,( zyxY

iiT 1

nT0

Orientation matrix

Position vector


Kinematics Equations

• Other representations– reference from, tool frame

– Roll-Pitch-Yaw representation for orientation

tooln

nref

toolref HTBT 0

0

,,, xyz RRRT

1000

0100

00

00

CS

SC

1000

00

0010

00

CS

SC

1000

00

00

0001

CS

SC


Solving forward kinematics

• Forward kinematics

z

y

x

p

p

p

6

5

4

3

2

1

1000zzzz

yyyy

xxxx

pasn

pasn

pasn

T

• Transformation Matrix


Solving forward kinematics• Roll-Pitch-Yaw representation for orientation

1000

0z

y

x

n

pCCSCS

pSCCSSCCSSSCS

pSSCSCCSSSCCC

T

1000

0zzzz

yyyy

xxxx

n

pasn

pasn

pasn

T

)(sin 1zn

)cos

(cos 1

za

)cos

(cos 1

xn

Problem? Solution is inconsistent and ill-conditioned!!


atan2(y,x)

x

y

yandxfor

yandxfor

yandxfor

yandxfor

xya

090

90180

18090

900

),(2tan


Roll-Pitch-Yaw 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


Roll-Pitch-Yaw Representation

,,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)


Roll-Pitch-Yaw Representation

0cossin yx nn

sin

cossincos

z

yx

n

nn

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


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


Example

Joint i i ai di

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

X33O

2O1O0O

Z2

Joint 1

Joint 2

Joint 3


ExampleJoint i

i ai di i

1 0 a0 0 0

2 -90 a1 0 1

3 0 0 d2 2

1000

0100

0cosθsinθ

0sinθcosθ

00

00

000

00

1 sin

cos

a

a

T

1000

000

sinθ

cosθ

1

1

1 1

sincos0

cossin0

111

111

2

a

a

T

))()(( 2103213

0 TTTT

1000

0sinθcosθ 22

22

223

100

00cossin

0

dT

1000

01iii

iiiiiii

iiiiiii

ii dCS

SaCSCCS

CaSSSCC

T


Example: Puma 560


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


Example: 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)l2

l1

l2

l1


Inverse Kinematics

6

5

4

3

2

1

65

54

43

32

21

10

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)


Thank you!

x

yz

x

yz

x

yz

x

z

y

Homework 2 posted on the web.

Due: Sept. 23, 2003

Next class: Jocobian Matrix, Trajectory planning

city college of new york 1 jizhong xiao department of electrical engineering city college of new...

Documents

rotation slide

oxyz slide

w new frame

robotics slide

plate slide

resultant rotation matrix

rotation submatrix

x y z slide