3 Mechanical systems

3.1 pose description and transformation

3.1.1 Description of coordinate system

Example 1

0x

0y

1x1y

)/(cos

kinematics Inverse

sin

cos

kinematics Forward

01

0

0

lx

ly

lx

l

Preliminary

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

z

P

u

vw

O, O’

Point represented in OXYZ:

zwyvxu pppppp

Tzyxxyz pppP ],,[

Point represented in O’uvw:

Two frames coincide ==>

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

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 , , 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

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

),(

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

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

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!

Example 2 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

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

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}

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

10

0

31

13BA

BA RT

10 31

'33

oA

BA rIT

Example 3 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

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

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’)

Homogeneous Transformation

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 ),,(

21

10

20 AAA

Composite Homogeneous Transformation Matrix

0x

0z

0y

10 A

21A

1x

1z

1y 2x

2z2y

?i

i A1Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices

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

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

x

y

z

u'

v'

v "

w"

w'=

=u"

v'"

u'"

w'"=

Euler Angle I, Animated

Euler Angle I

100

0cossin

0sincos

,

cossin0

sincos0

001

,

100

0cossin

0sincos

''

'

w

uz

R

RR

Euler Angle I

cossincossinsin

sincos

coscoscos

sinsin

cossincos

cossin

sinsincoscossin

sincos

cossinsin

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, .

Matrix with Euler Angle II

cossinsinsincos

sinsin

coscossin

coscos

coscossin

sincos

sincoscoscossin

cossin

coscoscos

sinsin

Quiz: How to get this matrix ?

Description of Roll Pitch Yaw

X

Y

Z

Quiz: How to get rotation matrix ?

3.2 Transmission mechanisms

Mechanisms are motion converters in that they

transform motion from one form to some required form.

Machine is a system that transmits or modifies the

action of a force or torque to do useful work. (involving

transmitting motion and force or torque)

Kinematics is the term used for the study of motion

without regard to forces.

3.2.1 kinematic chains

Link is a part of a mechanism which has motion relative

to some other part.

Joint is the point of attachment to other link which allows

the links attached to have relative movements among each

other.

Kinematic chain is a sequence of joints and links.

3.2.2 The four-bar chain

3.2.3 The slider-crank mechanism

The slider-crank mechanism is an extremely cost-effective means of converting rotary to linear motion.

The crank portion is the wheel that rotates about its center and has a rod of fixed lenght mounted to a point on its circumference; the other end of the connecting rod is attached to a linear stage which is constrained to move in only one dimension on a relatively frictionless surface.

At both its location the connecting rod is free to rotate thus the angle formed with the horizontal will change as a function of the disk’s position.

As the disk travels from 0 to 180° in the counterclockwise direction, the linear stage moves a distance equal to 2r: if the disk continues to travle from 180° back to 0° - still in counterclockwise direction, the load will move in the opposite direction over exactly the same linear distance.

3.2.4 cams

Law of motion for the follower: position, velocity, acceleration and jerk

Machining problems: undercut

Pressure angle

Cams – design criteria

3.2.5 Gear trains

Gear Train Ratio

The train ratio of a gear train is the ratio of the angular velocities of input and output members in the gear train. The train ratio here includes two factors, the magnitude and the relative rotating direction of the two members.

Advantages

Gear mechanisms are widely used in variety fields. They can be used to transmit motion and power between two any spatial rotating shafts and there are many advantages for them, such as, big range power, high transmission efficiency, exact transmission ratio, long life and reliable performance etc . .

Harmonic Drive Harmonic Drive is

a mechanical device for transmission of motion and power, which is based upon a unique principle of controlled elastic deformations of some thin-walled elements.　

Components

The three basic components of a Harmonic Drive are:

wave generator, flexspline and circular spline. Any

one of the three components may be fixed, while one

of the remaining two may be the driver, and the other

the driven, to effect speed increase or speed

decrease and at fixed speed ratio. Or, two of the three

components may be the driver while the third the

driven so as to effect differential transmission.

Characteristc High reduction ratio with wide range High precision Small backlash large torque capacity High efficiency Small size and light weight Smooth running Low noise

3.2.6 Ball screws 　

Ball screw assembly is consisted of screw, nut

and ball. The function is transfer the rotary motion into linear motion or transfer the linear motion into rotary motion.

We have a great variety of ball screw at high performance, cost effective, and extensive for machine tool, production machinery, precision instrument, promote the CNC development of machine tools thoroughly. Our products have the great feature of long operating life, high accuracy to C3 and C5, high transfer efficiency and good synchronization, fast delivery on many models.

Characteristics

• high transfer efficiency• high positioning accuracy• reversibility • long service life • good synchronization

F—0

F—F

F—F

J—J

Installation modes

3.2.7 Belt and chain drives

3.3 Oriented mechanisms

Linear oriented Guider

Configuration combination

3.4 Execute mechanisms

Mechanical Equipment

Electrical Equipment

Hydraulic Equipment

homework: page 115, problem 5,8