the city college of new york 1 kinematics of robot manipulator introduction to robotics

The City College of New York

1

Kinematics of Robot Manipulator

Introduction to ROBOTICS


2

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


3

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 objects3) Re-programmable: can do different things4) 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


4

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


Robotic Manipulators

5

• a robotic manipulator is a kinematic chain– i.e. an assembly of pairs of rigid bodies that

can move respect to one another via a mechanical constraint

• the rigid bodies are called links

• the mechanical constraints are called joints


A150 Robotic Arm

6

link 3link 2


Joints

7

• most manipulator joints are one of two types

1. revolute (or rotary)– like a hinge– allows relative rotation about a fixed axis between two links

• axis of rotation is the z axis by convention

2. prismatic (or linear)– like a piston– allows relative translation along a fixed axis between two links

• axis of translation is the z axis by convention• our convention: joint i connects link i – 1 to link i

– when joint i is actuated, link i moves


Joint Variables

8

• revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable

• qi : joint variable for joint i

1. revolute

– qi = i : angle of rotation of link i relative to link i – 1

2. prismatic

– qi = di : displacement of link i relative to link i – 1


Revolute Joint Variable

9

• revolute– qi = i : angle of rotation of link i relative to link

i – 1

link i – 1

link i

i


Prismatic Joint Variable

10

• prismatic– qi = di : displacement of link i relative to link i –

1 link i – 1 link i

di


Common Manipulator Arrangments

11

• most industrial manipulators have six or fewer joints– the first three joints are the arm– the remaining joints are the wrist

• it is common to describe such manipulators using the joints of the arm– R: revolute joint– P: prismatic joint


Articulated Manipulator

12

• RRR (first three joints are all revolute)• joint axes

– z0 : waist

– z1 : shoulder (perpendicular to z0)

– z2 : elbow (parallel to z1)

z0z1 z2

waist

shoulder

elbow

forearm

1

2 3


Spherical Manipulator

13

• RRP

• Stanford arm – http://infolab.stanford.edu/pub/voy/museum/pictures/display/robots/IMG_2404ArmFrontPeek

ingOut.JPG

z0z1

z2

waist

shoulder

1

2

d3


SCARA Manipulator

14

• RRP

• Selective Compliant Articulated Robot for Assembly

– http://www.robots.epson.com/products/g-series.htm

z0

z1 z2

1

2

d3


15

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


16

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


17

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?

how accurately a specified point can be reached

how accurately the same position can be reached if the motion is repeated many times


18

What is Kinematics

• Forward kinematics

Given joint variables

End-effector position and orientation, -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyxY x

y

z


19

What is Kinematics• Inverse kinematics

End effector position

and orientation

Joint variables -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyx

x

y

z


20

Example 1

0x

0y

1x1y

)/(cos

kinematics Inverse

sin

cos

kinematics Forward

11

1

1

lx

ly

lx

l


Example II

21

• given the joint variables and dimensions of the links what is the position and orientation of the end effector?

2

1

a1

a2


Forward Kinematics

22

• choose the base coordinate frame of the robot– we want (x, y) to be expressed in this frame

2

1

a1

a2

(x, y) ?

x0

y0


Forward Kinematics

23

• notice that link 1 moves in a circle centered on the base frame origin

2

1

a1

a2

(x, y) ?

x0

y0

( a1 cos 1 , a1 sin 1 )


Forward Kinematics

24

• choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame

2

1

a1

a2

(x, y) ?

x0

y0

( a1 cos 1 , a1 sin 1 )

1

x1

y1


Forward Kinematics

25

• notice that link 2 moves in a circle centered on frame 1

2

1

a1

a2

(x, y) ?

x0

y0

( a1 cos 1 , a1 sin 1 )

1

x1

y1

( a2 cos (1 + 2), a2 sin (1 + 2) )


Forward Kinematics

26

• because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame

2

1

a1

a2

x0

y0

( a1 cos 1 , a1 sin 1 )

1

x1

y1

( a2 cos (1 + 2), a2 sin (1 + 2) )

(a1 cos 1 + a2 cos (1 + 2), a1 sin 1 + a2 sin (1 + 2) )


Forward Kinematics

27

• we also want the orientation of frame 2 with respect to the base frame

– x2 and y2 expressed in termsof x0 and y0

2

1

a1

a2

x0

y0

1

x2y2


Forward Kinematics

28

2

1

a1

a2

x0

y0

1

x2 = (cos (1 + 2), sin (1 + 2) )

y2 = (-sin (1 + 2), cos (1 + 2) )

x2y2


Inverse Kinematics

29

• given the position (and possiblythe orientation) of the endeffector, and the dimensionsof the links, what are the jointvariables?

2 ?

1 ?

a1

a2

x0

y0

x2y2

(x, y)


Inverse Kinematics

30

• harder than forward kinematics because there is often more than one possible solution

a1

a2

x0

y0

(x, y)


Inverse Kinematics

31

law of cosines

2 ?

a1

a2

x0

y0

(x, y)

22221

22

21

2 )cos(2 yxaaaab

b


Inverse Kinematics

32

21

22

21

22

2 2)cos(

aa

aayx

)cos()cos( 22

and we have the trigonometric identity

221

22

21

22

2 2cos C

aa

aayx

therefore,

We could take the inverse cosine, but this gives only one of the two solutions.


Inverse Kinematics

33

1cossin 222

to obtain

Instead, use the two trigonometric identities:

cos

sintan

2

221

2

1tan

C

C

which yields both solutions for 2 . In many programming languages you would use thefour quadrant inverse tangent function atan2

c2 = (x*x + y*y – a1*a1 – a2*a2) / (2*a1*a2);s2 = sqrt(1 – c2*c2);theta21 = atan2(s2, c2);theta22 = atan2(-s2, c2);


Inverse Kinematics

34

221

22111 cos

sintantan

aa

a

x

y


35

Preliminary

• Robot Reference Frames– World frame– Joint frame– Tool frame

x

yz

x

z

y

W R

PT


Points and Vectors

36

• point : a location in space

• vector : magnitude (length) and direction between two points

p

qv


Coordinate Frames

37

• choosing a frame (a point and two perpendicular vectors of unit length) allows us to assign coordinates

0p

0q

1

2000 qpv0x

0y

0o 0

5.1

20q

5.2

40p


Coordinate Frames

38

• the coordinates change depending on the choice of frame

1p

1q

2

1111 qpv

1x

1y1o 1

2

5.01q

4

5.01p


39

Preliminary• Coordinate Transformation

– Reference coordinate frame OXYZ

– Body-attached frame O’uvw

wvu kji wvuuvw pppP

zyx kji zyxxyz pppP

x

y

z

P

u

vw

O, O’

Point represented in OXYZ:

zwyvxu pppppp

Tzyxxyz pppP ],,[

Point represented in O’uvw:

Two frames coincide ==>


40

Preliminary

• 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 yx y


41

Preliminary

• 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?


42

Preliminary• 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


43

Preliminary• 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

wP

u

CS

SCxRot

0

0

001

),(


44

Preliminary• 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

wP

u


45

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


46

Preliminary• Basic Rotation Matrix

– Obtain the coordinate of from the coordinate of

uvwxyz RPP

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

R

xyzuvw QPP

TRRQ 1

31 IRRRRQR T

uvwP

xyzP

<== 3X3 identity matrix

z

y

x

w

v

u

p

p

p

p

p

p

zwywxw

zvyvxv

zuyuxu

kkjkik

kjjjij

kijiii

Dot products are commutative!


Properties of Rotation Matrices

04/20/2347

• RT = R-1

• the columns of R are mutually orthogonal

• each column of R is a unit vector• det R = 1 (the determinant is equal to 1)


48

Example

• 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


49

Example• 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


50

Composite Rotation Matrix

• 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


51

Example• 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


Translation

52

• suppose we are given o1 expressed in {0}

1x

1y

1o

10x

0y

0o

0

0

301o


Translation 1

53

• the location of {1} expressed in {0}

1x

1y

1o

10x

0y

0o

0

0

3

0

0

0

300

01

01 ood


Translation 1

54

1. the translation vector can be interpreted as the location of frame {j} expressed in frame {i}

ijd


Translation 2

55

• p1 expressed in {0}

1x

1y

1o

10x

0y

0o

0

1

4

1

1

0

3101

0 pdp

1

11p

a point expressedin frame {1}


Translation 2

56

2. the translation vector can be interpreted as a coordinate transformation of a point from frame {j} to frame {i}

ijd


Translation 3

57

• q0 expressed in {0}

0x

0y

0o

0

1

2

1

1

0

300 pdq

1

10p

1

20q


Translation 3

58

3. the translation vector can be interpreted as an operator that takes a point and moves it to a new point in the same frame

d


59

Coordinate Transformations• 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}


60

Coordinate Transformations• Two Special Cases

1. Translation only– Axes of {B} and {A} are

parallel

2. Rotation only– Origins of {B} and {A}

are coincident

1BAR

ABB

AA'oPP rrRr

0r A'o


61

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

• Homogeneous transformation matrix

ABB

AA'oPP rrRr

1

r

10

rR

1

r BP

31

A'oB

AAP

10

PR

10

rRT 1333

31

A'oB

A

BA

Position vector

Rotation matrix

Scaling


62

Homogeneous Transformation• Special cases

1. Translation

2. Rotation

10

0

31

13BA

BA RT

10

rIT

31

A33

BA 'o


63

Example • 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

vw

O, O’hx

y

z

P

u

vw

O, O’


64

Example • Rotation about the X-axis by

1000

00

00

0001

),(

CS

SCxRot

x

z

y

v

wP

u

11000

00

00

0001

1w

v

u

p

p

p

CS

SC

z

y

x


65

Homogeneous Transformation

• 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


66

Example• 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


67

Homogeneous Representation• A frame in space (Geometric

Interpretation)

x

y

z),,( zyx pppP

1000zzzz

yyyy

xxxx

pasn

pasn

pasn

F

n

sa

101333 PR

F

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

(X’)

(y’)(z’)


68


• 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 ),,(


69


21

10

20 AAA

Composite Homogeneous Transformation Matrix

0x

0z

0y

10 A

21A

1x

1z

1y 2x

2z2y

?i

i A1 Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices


70

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

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

1000zzzz

yyyy

xxxx

pasn

pasn

pasn

F

ii A1

iA0

0x 0y

0z

a

b

c

d

e

1x

1y

1z

2z2x

2y

3y3x

3z

4z

4y4x

5x5y

5z

1000

010

100

0001

10

da

ceA

1000

0100

001

010

20 ce

b

A

1000

0001

100

010

21 da

b

A

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


71

Orientation Representation

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

• Any easy way?

101333 PR

F

Euler Angles Representation


72

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

Rotations about OW axis about OW axis about OZ axis


73

x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Euler Angle I, Animated


74

Orientation Representation• Euler Angle I

100

0cossin

0sincos

,

cossin0

sincos0

001

,

100

0cossin

0sincos

''

'

w

uz

R

RR


75

Euler Angle I

cossincossinsin

sincos

coscoscos

sinsin

cossincos

cossin

sinsincoscossin

sincos

cossinsin

coscos

''' wuz RRRR

Resultant eulerian rotation matrix:


76

Euler Angle II, Animated

x

y

z

u'

v'

=v"

w"

w'=

u"

v"'

u"'

w"'=

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


77

Orientation Representation

• Matrix with Euler Angle II

cossinsinsincos

sinsin

coscossin

coscos

coscossin

sincos

sincoscoscossin

cossin

coscoscos

sinsin

Quiz: How to get this matrix ?


78

Orientation Representation• Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?


Inverse Transformation

79

• the inverse of a transformation undoes the original transformation– if

– then

1000

dRT

10001 dRR

TTT


80

Thank you!

x

yz

x

yz

x

yz

x

z

y

Homework 1 is posted on the web.

Next class: kinematics II

the city college of new york 1 kinematics of robot manipulator introduction to robotics

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