The Kinema Tic Synthesis of Steering Mechanisms

8/19/2019 The Kinema Tic Synthesis of Steering Mechanisms

1/21

The

Kinematic

Synthesis

of

Steering Mechanisms

Jin

Yao

Departnwnt of

Mechanical

Engineering

Sichuan Union University

Chengdu

P.

R.

China

Jorge Angeles

Department of

Mechanical

Engineering

Centre for Intelligent

Machines

McGill

University

817

Sherbrooke

Street

Montreal

Canada

H3A 2K6

angeles®cim mcgill. ca

Abstract

We

propose

a computational kinematics

approach

based

on elimination

procedures

to

syn

thesize

a

steering four bar linkage.

In

this regard

we aim

at minimizing

the

root mean

square

error

of

the synthesized linkage in

meeting

the

steering

condition

over

a

number

of

linkage

con

figurations

within the

linkage range

of

motion.

A

minimization

problem

is thus

formulated

whose

normality

conditions lead

to

two

polynomial

equations

in

two

unknown design

variables.

Upon eliminating one of these two variables

a

monovariate polynomial equation is

obtained

whose roots

yield all

locally optimum linkages.

From these

roots the

global

optimum

as well

as

unfeasible

local

optima are readily

identified.

The

global optimum however turns

out to

be

impractical

because of

the

large differences in its link lengths which

we

refer

to as dimensional

unbalance

To

cope

with this drawback

we

use a

kinematically equivalent

focal

mechanism

i.e.

a six bar linkage with

an

input output function identical to that of the four bar linkage.

Given

that

the synthesized linkage requires

a rotational

input as opposed to

most existing steering

linkages

which

require

a

translational

input we propose

a

spherical

four bar

linkage

to

drive

the

steering linkage.

The

spherical

linkage

is

synthesized

so as to yield

a

speed reduction

as

close as

possible

to

and

to

have

a maximum transmission

quality.

La

synthèse cinématique

des mécanismes de direction

Résumé

Nous

proposons ici

une

approche

de

cinématique algorithmique basée

sur

une

méthode

d’élimination

de

variables qui permet de

synthétiser

un mécanisme de

direction

a quatre barres

articulées. Dans

cette

optique

nous

nous

efforçons

de

minimiser l’erreur quadratique moyenne

dii

mécanisme

obtenu en

vérifiant

la

condition

de

direction dans

urn

grand nombre

de

configura

tions possibles

pour

le

mécanisme

ce

qui mène

a

un probléme

de

minimisation.

Les

conditions


2/21

de

‘normalité I cc

dernier donnent lieu a

deux

equations

polynCmiales

a

deux variables

de con

ception

inconnues. L’élimination

de

l’une

de

ces

variables

permet

dobtenir

une seule

equation

polynômiale

a une

seule variable dont

les

racines

rCelles

donnent

tons

les minimums

beaux

Ii

es t

done possible

’en

tirer Ic

minimum global

et de

distinguer

les

minimums inadmissibles

‘ependant. Ic minimum

global

comporte des

longueurs

de

maillons

très

diverses

cc

qui Ic rend

pen

utile. NCanmoins le

concept

de rnécanisme focal

equivalent

a

savoir

un mCcanisme

a

six barres

articulCcs

ayant la

mCme relation entrée-sortie

que le

mCcanisme a quatre barres

permet de remplacer

cc

dernier

par

un

mécanisme

de

proportions

plus

acceptables. Puisque

Ic

mCcanisme

ainsi synthCtisC exige un cut rainement par couple rotoIde.

contrairement

aux

mCcanismes

de

direction

existants qui

exigent

un

entraInement

par couple glissant

nous pro

posons

un

mCcanisme sphérique pour coupler le mouvement

de

l’arbre

de

direction

au

maillon

entrainant du mCcanisme

de direction.

Le mCcanisme

sphCrique en question

est

synthétis

afin

de

donner une relation

entré-sortie

Cquivalente

a

une

transmission par

engrenages

a

reduction

2:

et

une qualitC

de

transmission

optimale.


3/21


4/21

synthesize

the

spherical

four—bar

linkage, whose

geometry

angles

between neighbouring

axes

is

then

determined

via

a

least-square

procedure

as

well.

2 Kinematic

Synthesis

of

the Steering

Four-Bar

Linkage

Within

the

kinematic synthesis

of

a

steering four-bar

linkage, there is

not

really an

input and

an

output

link,

but

the

two links must

move in a

coordinated fashion

so

as

to comply with

the steer?Tig

condthon.

1nder this condition.

the

two

wheel—carrying

links

are

required

to

move

in

such

a

way

that

the

two

axes

of the wheels.

£

and

M.

of

Fig. 1.

intersect the

rear axis

Al

of

the

vehicle.

Therefore.

these two

links

although

remaining

close to

parallel

upon steering

rotate about

their

vertical

axes

by differing

angles While

this difference is

relatively

small

it is

large enough

to warrant careful

consideration

when designing

the

mechanism.

Moreover, a

steering four-bar

linkage is required to

be

symmetric

since

it turns the

vehicle

both right and

left.

Figure

The wheel-steering

condition

Figure

1,

based

on one given

in

[6],

depicts the

configuration

of

the

wheels under

a

left turn.

According with the steering condition the

axes of

the

two

wheels

must intersect the rear

axis

at

a

common point I. From

that figure the

steering condition is readily obtained.

Indeed

from triangles

ME

and

DIE

we have

1

K

I

E

R

whence

b E

a

tan

DE

a tani9

b

tan

t

a

2

which

is

the

steering

condition

sought.


5/21

2.1

The

Set

of

Equations

for

Steering-Linkage

Synthesis

The

steering

condition

of eq.

2 is numerically

unsuitable

because

it

includes

variables

that

can attain

unbounded

values.

A

numerically more

convenient form of

tins

condition

will

be derived.

To

this

end,

we

redraw

Fig.

in

a

more

convenient

form, in

Fig.

2. with the

purpose

of

deriving later

the

input-output equation

of

the

steering

four—bar

linkage in a

more

familiar

layout.

4

D

Figure 2:

Notation

on

the

steering

condition

Now we

write the

steering condition

in terms

of

angles

cii

and

ci

illustrated in

Figs.

I and 2,

thus

obtaining

sin ci

psin

sin

ci2

0

3a

with parameter

p

defined as

p

3b

Equation 3a

is numerically

better behaved

than

eq. 2 ,

for

it

is expressed

in

terms

of

quantities

that

vary

between —

and +1.

Moreover,

the

input-output

equation

of the four-bar

linkage

at

hand

is

written in the

standard form

of

the

Freudenstein equation

[19],

in

terms

of

angles o and

o ,

indicated

in Fig.

3.

This equation

is

k

+

kosa

kos

a

cos ai

4

where

k

k and k

are

dimensionless parameters

defined

as

2

2 2

2

a

+

a

a

+

a

k

—

k—

a

a a

Note that

we

have a

04 , and

hence,

when

the

vehicle

travels along a

straight

course, the mid

planes

of the

two

front

wheels remain

parallel. This

leads

to 0

when

2

0,

while

eq. 4

leads

to

202

sin 3

6

Further.

we

use

the

relation between and

a

with

Q:

and

Q2.

which is

apparent

from

Fig. 3b:

a

=n/2— 3+Qi .

a /2

3—

ó

7


6/21

I

B

c

C

iDX

a

a)

Figure 3:

Four-bar linkage

with

the

standard

Freudenstein

notation in

a) a

straight-course configu

ration; b)

an

arbitrary configuration

Now,

let

x

and y

be

the nondimensional coordinates

of

point B

of Fig.

3b, i .e .,

x

=

sin/3 -sin/3 8a )

y =

cos 3

E

cos

8b)

a

It

is

apparent

from Fig.

3

that

neither

x

=

0

nor

y

= 0

can

be accepted as viable solutions. Indeed,

if

y

=

0,

then

3

=

900,

which in the presence of

a

symmetric linkage

produces

a

parallelogram,

whose

input

and output links

sweep

identical angles

and cannot, therefore, generate the steering condition.

If,

in

turn, x

= 0,

then the

linkage becomes a

structure,

with

al l

its joint

centres

aligned.

In

light

of

definitions

8a )

and

8b),

the Freudenstein equation becomes

f

Y

Ax

— Ay

+

Bx

+ Cy +

Dxy

= 0

9a)

where

A 1

—

cos cbi

—

B

2 — cos g1 —

cos 5

C

sin q1

—

sin i),

D

E

2sin q

q5

9b)

Thus, the

new

Freudenstein equation

includes

only

two

geometric

parameters,

and k or,

correspondingly,

x

and

y,

which are

to be

determined

so as to

produce

the

relation between angles

and

112

given

by

the steering condition.

By

virtue

of

the number

of

linkage parameters,

however,

the symmetric four-bar

linkage

at hand cannot

meet

exactly the steering condition

3a ) at more

than

two pairs of

,

t/2

values This linkage however,

can

approximate

more

than

two

pairs of those

values with

a

minimum error if

3

and

k or , equivalently x

and

y,

are

chosen appropriately.

We

then

formulate the

synthesis task

under

discussion as an

optimization

problem:

in

zf mm

lOa)

where

f

denotes function

f 1,

;

x.

y).

defined in eq. 9a), when evaluated at the ith

pair of

,

2

values, and

n

is

the number of such

pairs.

i.e.,

i=1,2,.. . ,n lOb)


7/21

with

A,

B a

nd

C,

de

fined

as

the valu

es

att

ained

b

y

4

B,

a

nd

C.

resp

ecti

vely

,

a

t th

e

i

th

pair

of

Qi.

2

v

alue

s

In orde

r to

min

imiz

e

we

ob

tain

the

n

orma

lity

co

nditi

ons of th

e

unc

onst

rain

ed mi

nimi

z tio

n

prob

lem o

f

eq. l

Oa .

i

.e..

24

i’

B,

±

Dy

0

h

a

—2A

y

+

+

D

’x

0

h

lb

It

s

houl

d be n

oted that

th

e

norm

ality

co

nditi

ons are not

orio

us

for

thei

r

ill—conditioned

b

ehav

iour

[20], a

nd he

nce, uns

uitab

le

for a

pur

ely nu

meri

cal

ap

proa

ch; how

ever

we

w

ill

not

pur

sue he

re

s

uch

an a

ppro

ach. Inst

ead,

we

wil

l

res

ort to

e

limi

natio

n proc

edu

res,

wh

ereb

y

th

e n

orm

lity cond

ition

s

offer

the

advantage

of

providing

all

possible

solution

s.

By

the sam

e tok

en, a

n

ume

rica

l

solu

tion of

Pro

blem

lOa yield

s

o

nly one

loc

al

m

inim

um

a

t

a

time

,

t

he

g

loba

l

min

imum

neve

r bein

g

g

uara

nteed

.

ls

o

n

ote tha

t,

is

q

uadr

atic in x an

d

y,

the tw

o fore

goin

g

cond

itio

ns thu

s

bein

g

cub

ic in x

and

y,

and

he

nce,

c

an be ex

pres

sed

as

p

p

X

P3’

P4

0

12

q

q

q

q

0

13

wh

ere

p

and

q

a

re co

nstan

t,

a

ll othe

r coe

ffic

ients

b

eing

pol

ynom

ials

in

y

wit

h

P2

an

d

q

lin

ear,

p

a

nd

q

qu

adra

tic,

and

p

an

d

q4

c

ubic

.

Monovariate

Polynomial

I

n

o

rder

to

so l

ve

e

qs.

12

an

d 13 f

or

x

an

d

y

we

e

limi

nate

on

e

un

know

n fr

om

t

he abo

ve two

eq

uatio

ns,

thu

s de

rivin

g

a

sin

gle

m

ono

vari

ate

po

lyno

mia

l

in

y

who

se

ro

ots yield

all

so

lutio

ns.

We

ai

m here

at

eli

min

ating

x

To this

end

.

we

mu

ltipl

y the

t

wo n

orm

ality

cond

itio

ns

f

irst

b

y x

an

d

th

en

by x

t

here

by p

rodu

cing fou

r

add

ition

al poly

nom

ial

e

quat

ions

,

w

hich

,

tog

ethe

r

w

ith eqs

. 12

and

1

3 , y

ield

six

lin

ear

ho

mog

eneo

us

eq

uatio

ns

i

n

[xm x

x z

i

.e.,

P1 P2

P3 P4

0

0

.x

0

q

q q

q

0 0

x

0

m

P2 P3

P4

0

x 0

0

q

q

q

q

0

0

14

0

0

m P2 P3

P4

I

0

qi

q q

q4

1

0

Obv

ious

ly,

th

e

s

olut

ion

so

ught

can

not b

e

tri

vial,

for the la

st

com

pon

ent of the

unk

nown v

ecto

r

is un

ity. Th

e

c

ondi

tion

fo

r

a

nont

rivia

l sol

ution

to

e

xist

is

t

hat

th

e

d

eterm

ina

nt of

tile co

effic

ient

matr

ix

he

z

ero Upo

n

e

xpan

sion

.

th

e de

term

inan

t

turns

ou

t

t

o he

nom

c in

y

n

ame

ly.

g y

7

6

5

±

r3

+

r

r

ro

0

15

w

r

co

effic

ient

s

{ k

}

a

re

co

nsta

nt. Mo

reov

er, n

otic

e

t

hat

the

solu

tion

0

w

as

al

read

y

found

un

acce

ptab

le, and

hen

ce,

t

he

abo

ve

n

onic equa

tion red

uce

s

to

an

oc

tic

eq

uati

on, i.

e.,

+

7

r6

5

r

4

r3

ro

0

16


8/21

2.3

Solution

Procedure

After

solving for

y

from

the foregoing equation

and substituting

it

into

eq. 14 .

we

obtain a

linear

system

of

equations

in

the

powers

of

x

which

thus

ields one

unique

value

of

this variable

fo r

each

of

the

roots

of

eq i6 .

With

x and

y

known,

we obtain

k and

d:

k

arctan

17

x

Moreover,

once a

has been decided on,

based on

the

space available.

03 and

o.

are

determined

04

a

aos

3

18

thereby

completing

the synthesis

procedure.

2.4

Numerical Examples

We illustrate

the

foregoing

procedure with

two

examples

taken

from the

literature,

for

comparison

purposes.

2.4.1

Dudià’s

Linkage

[9]

Let vary from

—40.0°

to

30.89°,

qS

varying

correspondingly

from

—30.89°

to 40.0°,

with

n

40,

and

all

40

points uniformly

spaced

along the

axis.

Moreover,

we

set

p

0.46,

as in

[9],

and

set

a

rn.

Equations

12

and 13 become

now, with six

decimals

displayed,

O.153878x

+

8.964945

+

7.44537y x

±

—2O9.385578y

+

86.606y

+

123.884916 x

—2.48179y

+

8

34.551135y

0

2.48179x

+

—104.691

+

81.60599y x

+

9l 7

7.44537y

+

44.78154 x

+0.153878y

+

6

788567

+

3.177492y

0

In

pursuing

the

monovariate

polynomial

approach, eq. 16

becomes,

with

only

four

decimals

dis

played,

—0.0008y

+ 0.0084y

+ 0.1069y

0.1858y

+0.2130y

0.159y

+

0.O7l6y

0.0149

0

Upon

solving

the foregoing

equation

fo r

y,

we

obtain four

real roots:

1 9117

Th

1.5131. ?J

1 2568

y

1.166

which

are all nondimensional.

Now, we

substitute

the foregoing

values

of

y

into eq. 14 ,

thereby

obtaining,

correspondingly,

four real solutions

for

x:

1.2373,

x

1.9272,

x 3.5746,

z 0 4992


9/21

Tab

le

G

eom

etric

al

P

aram

eter

s of the

Lin

kage

s

linkage

m

rn

in

erms

deg

ree

1.0 2

.277

1

].4

745 3

2.91

° 0

.17

2

1 0

2.45

02

2.85

43 51

.86°

0.

08

1

0

3.7

891

6.

1491 70.

63°

0 0

1

1 1

0 1.

2681

0.0

016

23.7

67°

We t

hus f

ind

fou

r

poss

ible

lin

kage

s

pr

oduc

ing

the ste

ering

con

ditio

n w

ith loca

lly—

min

imum

er

ror.

Show

n

i

n Tab

le

are t

he

geom

etri

c

p

aram

eter

s

o

f thes

e link

ages

, usin

g

i

n.

w

ith th

e

corr

espo

ndin

g roo

t-me

an s

quar

e erro

r

s

However,

only

the

first

three

of

the

above solutions are

acceptable, the fourth solution

giving

an

e

xtrem

ely

sh

ort

coup

ler

lin k

,

wh i

ch

y

ield

s

virtu

ally

a

tria

ngle

,

r

ather t

han

a

qu

adri

later

al.

F

or

this

r

easo

n, the c

orre

spon

ding

root

-mea

n s

quar

e error

has not

bee

n com

put

ed, th

e asso

ciate

d

en

try

in

T

abl

e bei

ng f

illed

w

ith

an

aste

risk

.

S

how

n

in

Fi

g. 4

are

the

pl

ots of th

e error di

strib

utio

ns

of

the

co

rres

pond

ing

linka

ges

.

Th

ese

h

ave

bet

ter

error

distr

ibut

ions

an

d

sm

alle

r

max

imu

m e

rror

s

ov

er

t

he

wh

ole st

eerin

g moti

on

than th

e linka

ge

obta

ined by D

udiä et

a

l.,

as

sho

wn

in F

ig .

5

It shou

ld

be

note

d t

hat

ha

s neg

ativ

e

len

gth

in

this

exa

mple

.

F

rom t

he

def

initi

on

of

eq.

6 ,

this

m

eans

that th

e

tw

o

lin k

s

A

B and

C

D ar

e

cr

ossed

whe

n t

he

f

our-b

ar

lin k

age is in

its

first po

sitio

n.

A

disc

ussio

n

on

how to

de

al wit

h neg

ativ

e

va

lues

of

or 4

in the

co

ntex

t of

li

nkag

e

syn

thesi

s for fun

ctio

n

gen

erat

ion

is

avail

able in

[19

]

2.4

.2

Fah

ey

and

Ho

usto

n’s L

ink

age [2

]

Th

e

se

cond

e

xamp

le is

that stud

ied

b

y

Fahe

y

and

Hu

ston [2

] in

whic

h

p

0

.6

an

d

th

e fu

ll

mo

tion

ran

ge is

[—

61

°, 4 1°

] Fo

llo w

ing the p

roce

dure

pro

pose

d

h

ere w

e obt

ain fo

ur rea

l

solu

tions

as

well

.

The

er

ror

pl

ot

o

f

the glo

bally

-opt

imu

m

so

lutio

n

is

s

how

n

in

Fi

g.

6

The resu

lt

show

s

tha

t th

e m

axim

um

str

uctu

ral

erro

r i

s

0.7

°

ov

er

t

he r

ange

[—

60° 0]

whi

ch is

much sm

aller

tha

n

th

at obt

aine

d

by

Fa

hey

a

nd

Hu

ston

.

o

f 2.93

°.

Rep

lace

men

t

o

f th

e Fo

ur

ar

Lin

kage w

ith

Its

F

oca

l

S

ix

ar

Equivalent

Al

thou

gh

t

he

f

our-b

ar ste

erin

g link

age

ob

taine

d ab o

ve satis

fies the

ste

ering c

ondi

tion

with hi

gh

ac

cura

cy

its larg

e d

imen

sion

al

unb

alan

ce

its l

arge

st

lin

k

i

s mor

e t

han si

x

time

s

long

er

than

its

s

hort

est

li

nk

ma

kes

it uns

uita

ble

f

or

prac

tica

l

ap

plic

ation

s.

A

s

an

alt

ernat

ive,

a

ki

nem

atica

lly

equi

vale

nt

six-

bar

lin

kag

e is de

rived

be lo

w from

the f

our-

bar lin

kag

e, with

a

bett

er

di

men

sion

al

b

alan

ce.

3

1

A Fou

r Ba

r L

inkag

e

an

d

ts

Foca

l

Six

Bar E

qui

vale

nt

Shown in Fig.

7

are

a

four-bar

lin kage

ABCD

and its

focal

six-bar

equivalent,

APFSDQF, in

wh

ich

and

stan

d

f

or the

ang

ular disp

lace

men

ts o

f

links AB

a

nd CD.

In ord

er

to let

l

ink s A

P and


10/21

nkage

/

HnKage 3

-Cl

—

-30

-20 -10

10

20

çb degree

Figure

4:

The error plots

for link g s 1,

2

and

3

of

Table

I

DQ

of

the six-bar

linkage

undergo

the same

angular displacements as AB and CD,

respectively, the

conditions shown below must be

observed [17]:

ab

a b

b d

19a

mn

+

pq

m

n

m +

n

—

p

—

q + d

and

S

mp

19b

d—s

nq

When

the dimensions

a, b c,

and d

of

the four-bar linkage r giv n

and

a

point

S

is

chosen along

line

AD,

a

distance

s

from point A,

we

can reduce

eqs. 19a and

19b

to a

quadratic

equation

in

A

With

A

known,

we

can calculate ni ,

n,

p,

and

q by

means

of

eq. 19a

and

2

— p

+

q

+

m

+

np

+

mnpq m

+

n

+p +

q

—

20

—

mp+nq

Fo r

the

first

example,

a =

b =

2.4502,

c

= 2.8543,

and d

= 1, with all lengths given in

meter;

taking

s

=

0.5 m,

we

have,

for

A =

15.2971,

m n

=

0 .1602 m,

p = q

=

0.6056

m

and

r

=

0.3774

m

for

the

focal six-bar

linkage,

as

shown in Fig.

8,

whereby it

is

apparent that

the

focal

six-bar linkage

is dimensionally

better

balanced

than

the original four-bar linkage.

3.2

General Form

of

the Focal Six-Bar

Linkage

With reference to

Fig.

9,

it

is

possible to

change the position

of

the

fixed

joint

S

of link SF

to 5’

by

a

combination

of

stretching and

rotation of

segment AS.

This

transformation

can

be

obtained,

e.g.,

as

[AS’ j{=LjjASj

21

s’= R s—a)+a

22


11/21

6

5[

4.-

0

JO

degree

Figure

5: The

error

plot

of

the

linkage

obtained

by

Dudiça

et

al.

[ ]

where R

is a

rotation

matrix

through

an angle

and

p

is

a

stretching factor.

Then,

the original

six-bar

linkage

PFS

QF

can be

transformed

into the

general

AP’F’F”Q’BS’

linkage. To

this

end,

the

condition

below

must be

satisfied [1,16]:

PF

S

AP’F’S’ and BQFS

BQ’F”S’

23

so

that

the

input-output

relation

between AP.

DQ

an d

AP’, DQ’

remains

unchanged.

When

p

and

p

are

given,

the general

six-bar

linkage

can

be

obtained

from the

position vectors

s’,

p’,

f’,

q’

and

f”,

of points

S’,

P’,

F’,

’

and

F”, respectively,

which

are

given

below:

s’

jtR s

a

+

a

24

p’=

pR p

—a

+ a

25

f’=p

R f—

a +

a

26

S’D

27

SD

q

’=nS

q—

d +

d

28

vS f

d

+

d

29

where S

is a rotation

matrix

through

an angle When

requiring

point

5”

to

he

located on

the

perpendicular bisector

of

A’D

and S’F’

S’F”

F’F”,

we

can

choose

pi

30°

to

do

so.

Tra

nsmi

ssio

n Qua

lity

In

Subsection

3.1.

when

a

different

s

is

chosen,

different

focal

six-bar

linkages

are obtained,

with

different

transmission

indices, and

SF

as

driving

link. The

transmission

index wa s

introduced by


12/21

1

o

5

5

I

ol

j

—8

—6

—4 —

4

6

deg

ree

Figu

re 6:

The

e

rror

cu rv

e

f

or the

glo

bal op

timu

m

of

the

seco

nd

ex

amp

le

Figure

7: Four bar

lin kage

and

its focal

six bar equivalent

Suth

erlan

d and

Ro

th [2 ] as

a ge

nera

lizat

ion

of t

he co

nce

pt o

f

t

rans

miss

ion ngl

e

In t

he

case of

plan

ar

f

our b

ar

lin

kage

s

the tr

ansm

issi

on index

re

duc

es

to

the si

ne

o

f

th

e

tra

nsm

issio

n an

gle . Ju

st

lik

e t

he tran

smi

ssion

angl

e whic

h

va

ries

with

th

e

con

figu

ratio

n

o

f

th

e

linka

ge

t

he

tra

nsm

issio

n

inde

x

i

s also co

nfigu

rati

on d

epen

dent

.

A glob

al p

erfo

rman

ce in

dex

m

easu

ring

the

go

odne

ss of

th

e f

orce

and

mo

tion

tr

ansm

issi

on

of the

lin

kage

is

the

tran

smis

sion quali

ty

[22

]

wh ic

h is def

ined in te

rms

o

f

Sut

herl

and and R

oths

tran

smis

sion

in

dex.

In

Fig.

10. we

di

sting

uish t

wo

fo

ur ba

r

lin

kag

es with a

com

mon i

nput li

nk .

SF.

of

u

t ut lin k

s

AP

and DQ.

In

that

figu re

link

SF

is

shown in

its

two

extreme positions

sweeping

the

whole

range


13/21

D

a

b

Figure

8: a

Steering

linkage

and

b its

focal

equivalent

Y

b

Figure

9:

The general

six bar focal linkage

A

1

P

S

X


14/21

of

m

otio

n.

S

an

d

SF

-.

L

inks

.4P

nd DQ

re disp

laye

d

ni

thei

r c

orre

spon

ding

e

xtrem

e

posi

tions

.

AP

and

l for

th

e

form

er.

ari

d

for

the latte

r.

Th

e

i

nput angl

e

i

s

in

dica

ted

l

ikew

ise,

f

or its

tw

o extr

eme

valu

es.

and

’

T

he tran

smi

ssion

ang

le

’

of th

e

l

eft fo

ur—

bar

link

age of

Fi

g. 1

is

know

n t

o b

e FP .

1.

while

its right

counterpart.

Pr

p

is

_FQD.

Let

Qi

and

stand

for

the t

ransm

iss

ion

qual

ity

of

the

left

and rig

ht

link

ages

, res

pecti

vely

,

i.e.,

i

’

s

in

t

=

sin

Th

e plo

t

of the

tra

nsm

issio

n

qu

alit

y

of

any

of the

tw

o

li

nkag

es

is

show

n in

Fig.

11 vs

. p

aram

ete

r

s.

d

efin

ed

in

Fig. ‘7

It

is appa

rent

fr

om

tins

plot that a

m

axim

um

v

alue o

f th

e t

rans

miss

ion qua

lity.

o

f

= Q

r

=

0.8

3

occ

urs

at

= 0.5

, a

re

sult

t

hat sh

ould be

e

xpec

ted b

y

s

ymm

etry

. H

enc

e,

in

o

rder

to

ach

ieve

a ma

ximu

m

tran

smis

sion

qua

lity,

link

S

F

sho

uld

h

e anc

hore

d

o

nto

th

e

veh

icle

ch

assi

s

half

way be

twe

en

t

he

anch

or po

ints

of th

e t

wo

o

utpu

t

l

inks AP

and

D

Q.

Bec

ause of

con

ditio

n 2

3 ,

the

g

ener

al f

ocal

six-

bar

lin

kag

e a

nd its

original

counterpart

have

both

the

sam

e tr

ansm

issi

on

quali

ty.

4

Th

e

Sy

nt

he

si

s

of

Sp

he

ric

a

l

F

ou

r

a

r

L

in

k

ag

e C

o

up

lin

g

t

he

S

te

eri

ng

W

he

el

w

i

th

t

he

S

te

eri

ng

M

ec

h

an

ism

Stee

ring

link

age

s

a

re

driv

en

by

t

he r

ot ti

on

of the

ste

eri n

g

wh

eel u

sing a

mec

hani

cal tr

ansm

issi

on

t

hat

i

s

base

d,

in mo

st

case

s,

on a

w

orm g

ear

[18

] Th

is

t

rans

miss

ion is s

uitab

le

to

the

cur

rent

desi

gn

of

st

eerin

g

lin

kag

es,

but

wo

uld

no

t b

e

s

uita

ble to

ou

r p

rop

osed

d

esig

n.

S

ince

t

he

d

rivin

g

link

of

our

desi

gn.

he

ncef

orth

r

efer

red

t

o

as

the

dri

ving

li

nk fo

r

bre

vity,

is

cou

pled to

the

vehi

cle

ch

assis

via a

rev

olut

e joi

nt,

the

mo

st

suit

able

tran

smis

sion

in this

c

ase

on

e t

hat

trans

mits

for

ce an

d

m

otio

n fro

m

th

e

st

eerin

g

whe

el,

wh

ose axis

is

inclined

with

respect to

the vertical

by

an angle

directly,

to

the

30

p

p

S

A O

X

\

/

F

F

Fig

ure 1

:

The

lim

it

posi

tions

of

lin

k

AP o

f the

foca

l

si

x-ba

r

link

age


15/21

055

0

01 02

03

4

05

06

7 08 09

Figure

11: Transmission

quality of

the steering

linkage

as a

function of

parameter

s of Fig. 7

driving

link of

the steering

linkage

whose

axis can be

assumed

to

be

vertical.

Furthermore

the axis

of the steering

wheel

is

offset

from

the

vehicle

midplane.

while

the axis

of the driving

link

is

located

in

this plane.

The

offset

can

be compensated

for

by means

of

a double

universal joint and

hence,

we

can

safely

assume

that

the

input motion to

the

steering

linkage is provided

via a

shaft

of

axis

intersecting

that

of

the

driving

link at the

above mentioned

angle The obvious

transmission

then

is

apparently

a spherical

four bar

linkage The

balance

of

this

section

is devoted

to the

synthesis of

this linkage

4

Des

cript

ion of

the

Veh

icle

ot t

ion

The

spherical

linkage

of

interest will

be synthesized

as a

function generator

[19],

which requires a

set

of

input output

pairs

of

values

that

the

linkage

is

to

meet.

In

order

to

define

the

input output pairs.

it

will

prove

convenient

to derive

a relation

between the

curvature

of

the vehicle

trajectory and

the

steering

angle

The

radius R

of the circular

trajectory

traced

by

the

vehicle

upon turning

is

often used

to

indicate

the

extent

of

the

vehicle

rotation. From

Fig.

2, R

can be

found

to

be

R

a

31

t

with

being

positive

when the

vehicle

turns to

the

right

and negative

otherwise.

the

same

holding

for

R.

When

o

tends

to

0, R

tends

to

infinity

and

hence

R

o

becomes discontinuous.

Fo r

this

reason

we prefer to

use

the

curvature

i

of

the

above

trajectory

which is

the

reciprocal of R,

i.e.,

2a btan

t nth

32

a

function

of

1i

that

remains

continuous

in

the

interval [h

where

K

K

K

K Q


16/21

with

t

and

o

defined

as

the two

extreme

values of

o:

in our

case.

=

—40

and

O

=

hown in

Fig.

12 is

a

plot

of

vs. i

which was obtained

using

ecj.

32).

Q,4

..Z.

03 02

0

01

Figure

12: Angle

vs .

curvature

i in

in

4.2 Input-Output

Function of the

Spherical Four-Bar

Linkage

A

spherical four-har

linkage

is

shown

in

Fig.

13,

in

which

TW is

the

fixed

link;

TU the input

link,

whose motion is identical

to

that

of

the

steering

wheel;

and

VW the output link, connected rigidly

to

FS of the steering six-bar

linkage The positions for the

input

and

output links are defined

by

o

+

t9

and

70

+

y,

respectively.

Fo r proper steering, the curvature

must

be

proportional

to the

input

angle i.e.,

k

= k9

33)

where

k

is a

scaling

factor. Therefore,

when k

and

k

are given, the

input displacement

t9

can be

obtained from

eq. 33 .

The

output displacement

‘y

in

turn,

is

obtained

by

imposing an input-output

relation. We

would like to

provide

a uniform

transmission

ratio

throughout

the

given

motion

range.

Additionally, a

torque amplification is required,

which

we

obtain

by

requiring

that the

transmission

ratio

be of

2:1,

but other transmission ratios

can

be

accommodated,

as

needed.

Fo r

a

value

of

k =

0.4313,

the plot

of

vs.

9

is shown in

Fig.

14.

We

discuss below the

synthesis

of

the spherical

linkage

using

an

optimization

approach.

4.3 Linkage

Optimization

The

input-output

function of

a

spherical

four-bar

linkage

is known

to he

[23]

F 9.

‘y

k

+

k cos 9o

+

?

+

k cos i9

+

i9)

cos 7

+

7)

—k

cos 7

+

7)

+

sin 7

+7

sin o

+

= 0

34)


17/21

F

igur

e

13:

The sph

eric

al f

our-

bar

lin

kag

e

w

here

c

os a cos

co

s

OS a

s

in

i

ll 4

S

ill

a

c

os

a

ill

a

k

c

os

a

sin c

os

a

sin

a

In sy

nthe

sizin

g

the

sp

heric

al

lin

kage

we

ass

ume

that

t

he

s

teeri

ng six

-bar

link

age

li

es in

a

h

oriz

onta

l

pla

ne

a

nd

t

hat the

axis

of

the

ste

erin

g-wh

eel ax le m

akes

45°

wi

th

t

he

verti

cal. H

enc

e

a

4

5°,

the

opti

miza

tion

proc

edur

e

thu

s

con

sistin

g

of d

eter

mini

ng

a

an

d

a

that p

rodu

ce

a

link

age

m

eetin

g

a

set o

f in

put-

outp

ut

p

airs with

a

min

imu

m

e

rror

.

W

e

a

im

at

the m

inim

izati

on

of

the Eu

clide

an

no

rm

of th

e des

ign error

ov

er

a

ric

h

se

t

of

in

put-o

utpu

t

pan

s,

n

amel

y

n

Z

E

3s

wh

ere

F t9

36

a

nd

{ i9

}

is

a

set

of n

po

ints

al

ong th

e

in

put-

outp

ut

f

unct

ion

F

’z9

‘

o

f

e

q. 34

and

disp

laye

d

in

F

ig. 1

4. Wh

en

r

an

d

are

kno

wn ,

the

fo

rego

ing

m

inim

izati

on redu

ces to so

lvin

g

a

li

near le

ast-

squ

are pro

blem

. Fo

r

ex

amp

le, wi

th r9 50°

an

d

7

o

—

80°

,

we

sp

ecif

y

40

pai

rs

of

equ

ally-

spac

ed

in

put-

outp

ut

v

alue

s

{ i

9.

‘

}

T

he re

sult

s

of

the o

ptin

uzat

ion are s

how

n in

T

able

2.

In

th

e

o

ptim

izati

on pro

ced

ure

we

impo

se the

con

ditio

n

t

hat

the

inpu

t link

b

e

a

cr

ank.

Thi

s ca

n

he r

eadi

ly

done

upo

n

re

calli

ng

th

e m

obil

ity crite

ria de

rived

by

Z

heng and

ng

eles

[ 3

:

k

+

k

k

k


18/21

4

2—

0.1—

—

6

—04 — 02

0 2

4

6

t9

rad

Figure

14:

The

input-output

function

of

the spherical

four-bar linkage

Table 2:

Optimum linkage with

io

= 500

and

=

—80°

k k k

ü

U

U

U

0.1334 2.019 6

0.7071

4.4809

45°

8.9675°

49.2807°

19.2961°

Now.

we regard

LI

and

o

as

additional

design variables to

minimize

the

sum

of the squares

of the

structural

error at

the

given input

values

The

optimum results

are

LI

= —1.4825°

o

=

—92.554°

the

other

parameters

being

shown in Table

3. The

output

link of

the

second

linkage

turns

out

to

be

a

crank as

well

with

a

transmission

quality

of

0.7689

which is still acceptable,

while the

error plot

is

shown

in

Fig.

16.

Note that the

second

linkage

offers

an error

which is

one

order

ofmagnitude

smaller

Table

3: Spherical

linkage

with

optimum

values of

and

‘y

k

k

Ic Ic

U

U

U

U

L 0.0991

0.0037

0.7071

2.2652 45° 17.3366° 88.1041°

89.6981°

than that of

the first

o ne . H ow ev er this

is done at

the expense

of

a

lower

transmission

quality.

In

practice.

such

a

small

error is futile,

for the tight tolerances

required to

implement

it increases

the

production

costs.

However the

transmission

quality

impacts 011

the life span

of

the

mechanism.

and

hence.

becomes a

more relevant

performance

index

In

summary.

the first

linkage is

preferred

over

the

second

one.


19/21

0015

001

0005

-0005k

-001

0015

2

—0 8

— 6

—0 4

—

2 0 0.2 0 4 0 6

0.8

t9

rad

Figure

15: Error

plot of the linkage of Table

5

Conclusions

A computational-kinematics

approach wa s

introduced to synthesize

a

steering four-bar linkage The

objective

of

this approach is to

minimize the root-mean square

value of the design error in the

steering condition.

‘W e

acknowledge

that

the minimization

of

the

design

error

is

not equivalent to

that

of the structural

error. Nevertheless,

we invoke a recent result

indicating that the two errors

become

closer

to each

other

as

the cardinalitv of the data set of prescribed

input-output pairs

in

the

synthesis of function

generators increases.

By

means

of an elimination procedure. a polynomial

equation

in

on e unknown

is

derived

from the associated

normality condition,

from w hi ch a ll local

optima

are computed

and, hence,

the

global

optimum can

he

readily

found.

Moreover

we synthesize

a

kinematically-equivalent

six-bar

focal mechanism,

to replace the steering

four-bar linkage, which

turns

out to

show

an

unacceptable

dimensional

unbalance. The

six-bar

equivalent

linkage shows

much

better-balanced

dimensions. Finally,

we

propose

an optimum

spherical

four-bar

linkage to

couple

the steering-wheel

axle

with

the steering mechanism.

In

fact,

two candidate mechanisms are

obtained,

with slightly different

performances. Based

on practical considerations,

the

mechanism

with the

higher

transmission quality,

if

with

a

lower accuracy,

is preferred.

Acknowledgements

The research

work

reported

here

wa s made

possible under Research Grant OGP0004 532

of the

Natural

Sciences

and

Engineering

Research Council NSERC

of Canada. Mr

Ya o would

like

to

thank

the China Scholarship

Program

for

providing

him

support

during

his

one-year visit

to the

Centre

for Intelligent Machines,

McGill

University

Montreal.

Canada.


20/21

x

10

2

T

0-

—05--

—04 02

02

0.4 0.6

0 8

rad

Figure

16: Error

plot

d cs

‘ygen

efe

ren

es

Dijksman,

E. A. ,

1997,

“Do the Front

Wheels

of

Your Car Really

Meet Ackermann’s

Principle

When Driving Through

a

Bend of

the Road”, Personal Communication.

2.

Fahey, S.

O’F.

and

Huston,

D.

R. ,

1997,

“A

Novel Automotive

Steering

Linkage”,

Journal

of

Mechanical

Design, Vol . 119 ,

pp.

481

481.

3.

Cole, D.

J.,

1997, “Analysis

of

the

Dynamic Force

in

the

Steering

Linkage of

Off-Road

Vehicles”,

mt

J.

of

Vehicle

Design Vol. 18,

No . 1,

pp .

83

99.

4.

Zanganeh,

K. E.. Angeles ,

and

Kecskemthv

A. .

1995.

“On

the

Optimum

Design

of

a

Steer

ing

Mechanism.”

Proc. Seventh

World

Congress

on

the Theory of Machines

and

Mechanisms,

Milan,

pp .

2524 2528.

5

Reister, D. B. and

Unseren,

M.

A.. 1993.

“Position and Constraint

Force

Control

of

a Vehicle

with

Two

or

More Steering

Driving

Wheels ” IEEE

Trans. on

Robotics

and

Automation, Vol. 9,

No.

6,

pp .

723

731.

6.

Dudiä,

F.

L.,

Diacoiiescu, D.

and

Gogu Gr..

1989, Mezanisme

Articulate.

Inventica

Jinemat

ica,

Editura

Technicà, Bucharest.

7.

Ardayflo,

D.

D.

and Qiao

D., 1987, “Analytical

Design

o f Seven Joint

Steering

Mechanisms”.

Mech.

a-nd Mach.

Theory. Vol.

22.

No.

4,

pp .

315

319.

8.

Miller. G.

II..

1986.

“The Effect

of Ackerman

sic Steering

Correction

Upon

Front Tire

Wear

of

Medium

Duty

Trucks,” SAE

Paper

No. 861975.


21/21

9

D

udià

,

F

.

L

an

d

A

lexa

ndru

,

P

.

1972 “Sy

stem

atis

ierun

g

der Le

nkm

echa

nisrn

en

der Ba

d

f hr

zeug

e’

K

oris

trukt

ion. Vol

24

p

p.

54

61

10 Dudiã.

F.

L

and Alexandru.

P

975

“Synthesis of

the

Seven-Joint

p ce

Mechanism

Used

in the

Stee

ring

S

stem

o

f

R

oad

Veh i

cles

”.

Pro

c.

F

ourt

h W

orld

Gongrcs 5

on th€ T

heo

ry

of

Ma

chin

es

and M

ech

ani

5ms

.

New

castl

e

U

pon

Tvn

e

197 5

p

p.

6

97

702

11 Dv

ali

H R

an

d

A

leks

ishv

ili.

N

1971

“D

esig

n

for

an Aut

omo

bile

S

teer

ing

G

ear

is a S

phe

rical

Fo

ur-B

ar Li

nkag

e”, Mec

h.

and

M

ach.

T

heo

ry No

6

p

p.

167

175

12 Go

ugh. V

F.. 19 56

“Pr

actic

al

T

ire R

esea

rch.

”

SAE

T

ran

sacti

ons.

Vol

64

pp 310—318

13 C

om

be.

W.

.

1818 “O

bse

rvat

ions

on

Ack

erm

ann’

s

Pat

ent Mo

vabl

e

Ax

les

fo

r Four—Wheeled

C

arria

ges,

”

P

rinte

d for

R

.Ac

kerm

ann

. 101 St

rand

, by

J. D

igge

ns.

St .

Ann

’s L

ane. Lond

on.

14

Wolf

W. A.

1959

“Anal

ytica

l

Design

of

an Ack

erm

ann

S

teeri

ng Lin

kag

e”, A

SM

E J

ourn

al

o

f

En

gine

erin

g

for

I

ndu

stry,

pp.

10 14

15 Hay

es M

J.

D. Par

sa,

<

an

d A

nge

les

J.

,

1999

“T

he

ef

fec t

of d

ata-s

et

card

inal

ity o

n the

des

ign and

struc

tura

l

er

rors of fou

r-ba

r

func

tion-

gene

rato

rs”,

P

roc. T

enth

Wor

ld Con

gr

es s

on

the

T

heo

ry

of

Ma

c/tin

es a

nd

Me

chan

isms

.

J

une 20—24

Ou

lu, Finla

nd, Vol

1

pp .

43

7

442

16 Dij

ksm

an, E A.

1976 “How

to

Re

plac

e

the

F

our-

Bar Cou

pler

M

otio

n

by

th

e

Cou

pler

Mo

tion of

a

Si

x-ba

r

M

ech

anis

m

W

hich Do

es Not

C

ont

ain

a

Par

allel

ogra

m”, R

evue Rou

main

e

de

s Sc

ienc

es

T

ech

iques

Sér

ie

de

Méc

aniq

ue

A

ppli

quée

,

Vol

21

pp.

359

370

17

Wunderlich, W.

1968 “On

Burmester’s

Focal

Mechanism and Hart’s

Straight-Line

Motion,”

Jou

rnal of

M

ech

anism

s. Vol

3

pp

.

79—86

18

Dür

r,

R.

an

d S

chie

hlen

. W

., 1998

“M

odel

s

fo

r Sim

ulat

ion o

f

Pow

er-S

teeri

ng

S

ystem

s,”

in

A

ng e

les

J. and

Za

khar

iev,

E

eds.

, NA

TO

A

SI Se

rie s

F

Vol

161

S

prin

ger-

Verl

ag,

He

idel

berg

,

p

p.

268—295

19

De

navi

t,

J.

and

Har

tenb

erg,

R

S 1964

K

inem

atic Synt

hesi

s

of Lin

kag

es,

Mc

Gra

w-H

ill

Bo

ok

Co

. N ew Yo

rk .

20 IKah

aner

D

Mo

ler

C

and

Nas

h,

5

1989 N

ume

rical

Me

thod

s

an

d

S

oftw

are

Pren

tice-

Hall

,

Inc., Englewood

Cliffs N ew

Jersey .

21 S

uthe

rlan

d,

G

a

nd Both

, B

.

1973

“A

Tran

smis

sion Inde

x

for

S

pati

al Mec

hani

sms”

, ASM

E

Jour

nal

of

E

ngin

eerin

g

for Indu

stry

, Vol

95

pp.

598—597

22

Go ss

elin

C

a

nd A

ng e

les

J.,

1986

“O

ptim

izat

ion

o

f

Plan

ar

and S

pher

ical

F

unct

ion

Ge

nera

tors

as

Min

imum

-D

efect

L

inka

ges .

” Mech

a

nd M

ach.

The

ory.

Vol 24

No 4

pp.

293—307

23

Zh en

g.

L and An

geles

.

J

..

1992 “L

east

-Squ

are

Op

timi

zatio

n

of

Pla

nar an

d Sp

heri

cal Fou

r

b

ar

Fu

nctio

n

Ge

nera

tors Und

er Mob

ility Con

stra

ints”

,

Jou

rnal

o

f

M

ech

anic

al

Des

ign

Vol

114

p

p.

569

573

The Kinema Tic Synthesis of Steering Mechanisms

Documents

Transcript of The Kinema Tic Synthesis of Steering Mechanisms