Kinematics of Robot

Manipulator

Examples of Kinematics Calculations

• Forward kinematics

Given joint variables

End-effector position and orientation,

-Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyxY x

y

z

Examples of Inverse Kinematics• Inverse kinematicsEnd effector position and orientation

Joint variables -Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyx

x

y

z

Example 1: one rotational link

0x

0y

1x1y

)/(cos

kinematics Inverse

sincos

kinematics Forward

01

0

0

lx

lylx

l

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

x

yz

x

z

y

W R

PT

Coordinate Transformation– Reference coordinate frame OXYZ– Body-attached frame O’uvw

wvu kji wvuuvw pppP

zyx kji zyxxyz pppP

x

y

zP

u

vw

O, O’

Point represented in OXYZ:

zwyvxu pppppp

Tzyxxyz pppP ],,[

Point represented in O’uvw:

Two frames coincide ==>

Properties: Dot Product

• Mutually perpendicular • Unit vectors

Properties of orthonormal coordinate frame

0

0

0

jk

ki

ji

1||

1||1||

k

ji

Let and be arbitrary vectors in and be the angle from to , then

3R

cosyxyx

x yx y

Coordinate Transformation

• 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?

Basic Rotation• 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

Basic Rotation Matrix• Basic Rotation Matrix

– Rotation about x-axis with

w

v

u

z

y

x

ppp

ppp

wzvzuz

wyvyuy

wxvxux

kkjkikkjjjijkijiii

x

z

y

v

wP

u

CSSCxRot

00

001),(

Is it True? Can we check?

• – Rotation about x axis with

cossin

sincos

cossin0sincos0001

wvz

wvy

ux

w

v

u

z

y

x

ppp

ppppp

ppp

ppp

x

z

y

v

wP

u

Basic Rotation Matrices– Rotation about x-axis with

– Rotation about y-axis with

– Rotation about z-axis withuvwxyz RPP

CSSCxRot

00

001),(

0

0100

),(

CS

SCyRot

10000

),(

CSSC

zRot

Matrix notation for rotations

• Basic Rotation Matrix

– Obtain the coordinate of from the coordinate of

z

y

x

w

v

u

ppp

ppp

zwywxw

zvyvxv

zuyuxu

kkjkikkjjjijkijiii

uvwxyz RPP

wzvzuz

wyvyuy

wxvxux

kkjkikkjjjijkijiii

R

xyzuvw QPP

TRRQ 1

31 IRRRRQR T

uvwP

xyzP

<== 3X3 identity matrix

Dot products are commutative!

Example 2: rotation of a point in a rotating frame

• 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

2964.4598.0

234

10005.0866.00866.05.0

)60,( uvwxyz azRota

Example 3: find a point of rotated system for point in coordinate system

• 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

2964.1

598.4

234

10005.0866.00866.05.0

)60,( xyzT

uvw azRota

• Reference coordinate frame OXYZ• Body-attached frame O’uvw

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

Example 4: finding a rotation matrix• 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

SSSCCSCCSSCSSCCCS

CSSSCCSCSSCC

CSSCCS

SC

uRotwRotIyRotR

00

001

10000

C0S-010

S0C

),(),(),( 3

Reference coordinate frame OXYZBody-attached frame O’uvw

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}

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

'oAPBB

APA rrRr

0' oAr

Homogeneous Representation• 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

Homogeneous Transformation• Special cases

1. Translation

2. Rotation

100

31

13BA

BA RT

10 31

'33

oA

BA rIT

Example 5: translation• Translation along Z-axis with h:

1000100

00100001

),(h

hzTrans

111000100

00100001

1hp

pp

ppp

hzyx

w

v

u

w

v

u

x

y

z P

u

vw

O, O’hx

y

zP

u

vw

O, O’

Example 6: rotation• Rotation about the X-axis by

1100000000001

1w

v

u

ppp

CSSC

zyx

100000000001

),(

CSSC

xRot

x

z

y

v

wP

u

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

Example 7: homogeneous transformation

• Find the homogeneous transformation matrix (T) for the following operations:

44,,,, ITTTTT xaxdzz

:

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

axis OXabout Rotation

Answer

100000000001

100001000010

001

1000100

00100001

100001000000

CSSC

a

dCSSC

Homogeneous Representation• A frame in space (Geometric

Interpretation)

x

y

z),,( zyx pppP

1000zzzz

yyyy

xxxx

pasnpasnpasn

F

n

sa

101333 PR

F

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

(X’)

(y’)(z’)

Homogeneous Transformation

• Translation

y

z

n

sa n

sa

1000

10001000100010001

zzzzz

yyyyy

xxxxx

zzzz

yyyy

xxxx

z

y

x

new

dpasndpasndpasn

pasnpasnpasn

ddd

F

oldzyxnew FdddTransF ),,(

Homogeneous Transformation

21

10

20 AAA

ii A1

Composite Homogeneous Transformation Matrix

0x

0z

0y1

0 A2

1A

1x

1z

1y 2x

2z2y

?Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices

Example 8: homogeneous transformation based on geometry

• For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5

1000zzzz

yyyy

xxxx

pasnpasnpasn

F

1000010100

0001

10

dace

A

ii A1

iA0

0x 0y

0z

a

b

c

d

e

1x

1y

1z

2z2x

2y

3y3x

3z

4z

4y4x

5x5y

5z

10000100

001010

20 ce

b

A

10000001

100010

21 da

b

A

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

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

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

Euler Angle I, Animated

x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Orientation Representation• Euler Angle I

1000cossin0sincos

,cossin0sincos0001

,1000cossin0sincos

''

'

w

uz

R

RR

Euler Angle I

cossincossinsin

sincoscoscoscos

sinsincossincos

cossin

sinsincoscossin

sincoscossinsin

coscos''' wuz RRRR

Resultant eulerian rotation matrix:

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, .

Orientation Representation

• Matrix with Euler Angle II

cossinsinsincos

sinsincoscossin

coscoscoscossin

sincos

sincoscoscossin

cossincoscoscos

sinsin

Quiz: How to get this matrix ?

Orientation Representation• Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?

The City College of New York

38

The City College of New York

39

Kinematics of Robot Manipulator

Jizhong XiaoDepartment of Electrical Engineering

City College of New Yorkjxiao@ccny.cuny.edu

Introduction to ROBOTICS