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Dimensional synthesis of independent suspension mechanism s 1143

R ~ chassis R

. _ 2 : 4

ndependent g raph

suspens ion ~ schem at ic

a ) b ) c )

NOMENCLATURE

O -

vertexepresents

u s p e n s i o n l in k

~ - e d g e re p r e s e n t s j o in t

~ ) - v e r t e x w i t h d r c l e r e p r e s e n t s c h a s s i s

n u m b e r s i n s c h e m a t i c andgraph

s h o w l in k - t o - v e r t s x

correspondence

R - r e v o l u t e j o i n t S - s p h e r i c a l

joint

Fig 3 Gra ph representation of independent suspension

spatial linkage shown in Fig. 3a. This graph-based representation of linkage structure was

introduced in the mid 1960s and a good account of it may be found in [2]. We examined 26

automobiles, and graphs representing their suspension structures are shown in Fig. 4. In each graph

the vertex C represents the chassis, the vertex W represents the wheel hub, and the edges labeled

R, P, S, C, and K represent revolute, prismatic, spherical, cylindric, and Hooke joints, respectively.

We found nine distinct graphs which are arranged in order of increasing structural complexity in

Fig. 4.

At the low end of the spectrum is the trailing arm suspension. It consists of a single

revolute joint connecting the chassis to the wheel hub. This is followed by the planar short-long

arm SLA) suspension which is essentially a four-bar linkage in which the chassis is the base

link and the wheel hub is the coupler. Next comes the planar MacPherson strut which is a

slider-crank linkage. This is followed by the Jaguar rear suspension which is a four-bar

linkage in which one of the cranks can rotate about its own axis and serves as the

motion transmitting member connecting the vehicle transmission to the wheel. The self-rotation

of this member is made possible by the Hooke and spherical joints at its ends. It performs this

motion transmission function in addition to being a suspension member. The next graph is

that o f the spatial SLA suspension. The outer circuit represents an RSSR linkage in which the wheel

hub is the coupler. The rotation o f the wheel hub about the axis defined by the two coupler S joints

is constrained by an additional link known as the tie-rod running from the chassis to the wheel

hub. This is followed by the graph of the spatial MacPherson strut. It is an RSPS linkage with

the wheel hub being the coupler. The rotation of the wheel hub about the two S joints in the RSPS

circuit is constrained by a tie-rod similar to that in the spatial SLA graph. The next graph shows

the structure of the Nissan multi-link front suspension. The outer circuit of the graph represents

a spatial five-bar linkage with four revolute joints and one spherical joint. In addition there is a

tie-rod connecting the chassis to the wheel hub to constrain rotations of the wheel hub as desired.

This is followed by the graph of the Nissan DARS rear suspension. It has four chains of links and

joints connecting the chassis to the wheel hub. The final graph is that of the Mercedes five-link

rear suspension. It has five chains of links and joints connecting the chassis to the wheel hub.

All of these graphs represent single-degree-of-freedom systems. From the tables in Fig. 4 it is

clear that the MacPherson strut and the SLA have been very popular over the past decade. In recent

times, however, designers have turned to more complex multi-link arrangements in order to achieve

good elastokinematic behavior of the suspension. In particular the last three graphs represent

suspensions in which unwanted steer motions of the wheel during accelerated motions and

cornering maneuvers are minimized. We believe that the evolution of these suspension mechanism

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1144 MadhusudanRaghavan

c @ o w

t r a i li n g a m d

a l m l , @ l l l l~ m n m

1 9 8 8 B M W 7 5 0 i L r )

1 9 8 8 O p e l S e n a t o r B r )

O p ~ v e = a r)

1 9 6 4 M e ~ o e d e s 3 8 0 S E r )

S t roanJ X T ( r )

i a d m r t - lo n g a r m

l ~ n m 0

1 9 ~ T o y o t a S u p r a r ) I

a m t

1 s i n J a ~ m r x J s t )

1 9 8 6 H o n d a P re l u d e f )

1988 H ond a P re lude {f , r)

1 9 8 8 H o n d a C iv ic G R X ~ f )

~ 4 M e r ~ d ~ SS 0S E r)

1 9 8 6 Ho nd a ~:-: - :- :, l )

1 9 8 8 T o y o ta 8 u r n f )

D v er -c a rm ra i nM li r i um e s

1 9 8 8 H o n d a C i v b C R X ~ ( r)

~ 9 8 6

N ond a A oGord ( r )

l l m = P l w m n s t r u t

t l l l n l r )

1 9 8 8 U n c o l n C o n t . r )

I ~ F o rd P r d ~ r )

1 9 8 S F o rd T - , m s r )

1 9 8 6 H o n d a P r e ll x le r )

1 9 8 7 T o y o l a C , J c a r )

1 9 8 7 T o y o la C a m p / r )

i d a l N l l r l O U l q l l r I

1988 J agua r X J 6 ( r ) I

t i e - r o d t i e - r o d

M a c l = l l w m n s t r u t

1 9 8 8 B M W 7 5 0 L ( I)

1 9 8 8 l . ~ : ~ n C o n t . ( I)

1988 M z da C 41~ l a ( l , r)

S u b l r u X T ( f )

1 9 8 6 H o n d a C iv i c C R X / S i f

o ~ S e n a o r B 9

o p e l v ~ t r a p )

1 9 8 9 F o r d P nobe ( f )

1 9 8 8 F o r d T a u r u s / S a b l e f )

1 9 84 M a n ~ 1 90 E (f)

1 9 8 4 M a z d a R X - 7 f )

I g 8 7 T o y o t a C e l ~ a ( f )

1 9 8 7 T o y o t a

C am ry ( t)

1 9 8 6 A c u m Im e g r a f )

A . d i 4 0 0 O O u a t u o f ,r )

1 9 8 8 F ~ U n o s d e ~ t a (f )

M~_,.__d~shi r a m (f)

1 9 8 9 N i s s a r 1 2 4 0 S X f )

i N la .mn mu l t i- l ink

1 9 8 9 N il s a n 8 0 0 Z X f ) I

N im m n D A R S

]

1 9 8 9 N i s u n 2 4 0 S X r ) I

1 9 8 @ N i s s i m 3 0 0 Z X r ) I

S S

I1 Id er ~ d u ve41nk

9 8 4 M e r c e d e s 1 9 0 E r ) J

Fig. 4. Survey of indep endent suspension kinem atic structure.

N O M B I ~ C L A T U R E

R - r e v o l u t o J o i n t

p pa~mmc ioim

s - s p h e d u d J o in t

K - H o o k e J o i n t

W - w h e e l h u b

C - c h a s s i s

f ) - f r o n t

r ) - r e a r

d e s i g n s h a s b e e n f o r t h e m o s t p a r t e m p i r i c a l . T h e r e i s n o g u a r a n t e e t h a t c h a s s i s e n g i n e e r s h a v e

o v e r t h e y e a r s i d e n t i fi e d a l l u s e a b l e s u s p e n s i o n m e c h a n i s m s . T h e r e f o r e t h i s s u r v e y ra i s e d t h e

f o l lo w i n g q u es t io n : W h a t o t h e r m e c h a n i s m s a r e p o t e n t i a l c a n d i d a te s f o r u s e a s i n d e p e n d e n t

s u s p e n s i o n s ? T h i s i s a n s w e r e d i n S e c t i o n 3 w h e r e w e u s e n u m b e r s y n t h e s i s t o s y s t e m a t i c a l l y

e n u m e r a t e a n a t la s o f i n d e p e n d e n t s u s p e n s io n c a n d i d a t e s .

3. A N A T L A S O F L I N K A G E S F O R I N D E P E N D E N T S U S P E N S I O N S

T h e d e s i g n c o n s t r a i n t s f o r t h e n u m b e r s y n t h e s i s a r e a s f o l l o w s .

i ) S i n g l e - d e g r e e - o f - f re e d o m l i n k a g e s

T h e e n u m e r a t i o n i s r e s t r ic t e d t o s i n g l e - d e g r ee - o f - fr e e d o m l in k a g e s . T h i s i s b e c a u s e a t y p i c a l

r e a r s u s p e n s i o n h a s j u s t o n e d e g r e e - o f- f re e d o m p e r m i t ti n g t h e u p - d o w n m o t i o n o f t h e w h e el .

T h e s te e r m o t i o n o f t h e w h e e l n e ce s s a ry i n f r o n t s u s p e n s io n s m a y b e o b t a i n e d b y a d d i n g a

s u i t a b l y l o c a t e d r e v o l u t e j o i n t t o s u c h a s i n g l e - d e g r e e - o f -f r e e d o m s u s p e n s i o n m e c h a n i s m .

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Dimensional synthesisof indep ende nt suspension mechanisms 1145

Fig . 5. In-pa rallel linkage.

i i) I n - p a r a l l e l l i n k a g e s

W e e n u m e r a t e o n l y f u l l y i n - p a r al l e l li n k a g e s , i. e. l i n k a g e s i n w h i c h t h e r e a r e s e v e r al c h a i n s o f

l i nks a nd j o in t s c onne c t ing t he c ha ss i s t o t he whe e l hub ( se e F ig . 5 ) . T h i s r e s t r i c t i on i s ne c e ssa r y

b e c a u s e s u s p e n s i o n l i n k a g e s m u s t w i t h s t a n d l a r g e f o r c e s a n d m o m e n t s d u r i n g a c c e l e r a t e d m o t i o n .

I n - p a r a l l e l l i n k a g e s a r e s t r u c t u r a l l y w e l l - s u i t e d f o r s u c h a p p l i c a t i o n s .

i i i ) J o i n t s e t

T h e jo i n t s e t use d i n t he e n um e r a t i on i s a s f o l l ows : r e vo lu t e ( R) , p r i sm a t i c ( P ) , c y l i ndr i c ( C) ,

H o o k e ( K ) , a n d s p h e r i c a l (S ) j o i n t s .

T h e b a s i s f o r t h e e n u m e r a t i o n i s t h e f o l l o w i n g w e l l - k n o w n r e s u lt d u e t o G r u b l e r :

m = 6(n - 1) - 5nR -- 5rip -- 4nc -- 4nK -- 3ns, (sp atia l l inkag es)

m = 3(n - - 1) - - 2nR - - 2rip , (pla na r l inkag es)

I n t h e a b o v e f o r m u l a e , m d e n o t e s t h e n u m b e r o f d e g r e e s - o f - f re e d o m o f t h e l in k a g e a n d n is th e

t o t a l n u m b e r o f l in k s . T h e s y m b o l s nR , n p, n c , nK , a n d n s a r e t h e n u m b e r o f R , P , C , K a n d S j o i n t s

in t he l i nka ge , r e spe c t i ve ly . S inc e t he e nu m e r a t i o n i s r e s t r i c t e d t o s i ng l e - de gr e e - of - f r e e do m li nka ge s ,

m i n t h e a b o v e f o r m u l a e is se t e q u a l t o 1. W e p r o c e e d b y c o m p u t i n g a l l p o s s i b le j o i n t p e r m u t a t i o n s

l e a d i n g t o v i a b l e s u s p e n s i o n l i n k a g e s f o r e a c h p e r m i s s ib l e i n - p a ra l l el l i n k a g e t o p o l o g y . T h e r e a r e

t w o s o l u t i o n s t o G r u b l e r ' s f o r m u l a r e p r e s e n t i n g l i n k a g e s i n w h i c h t h e c h a s s i s i s c o n n e c t e d t o t h e

w h e e l h u b b y a s i n g l e j o i n t . T h e s e s o l u t i o n s a r e s h o w n i n F i g . 6 u n d e r t h e c a t e g o r y s i n g l e - c h a i n

l i n k a g e s . O n e o f t h e g r a p h s i s t h e w e l l - k n o w n t r a i li n g a r m t y p e s u s p e n s i o n p r e v i o u s l y e n c o u n t e r e d

i n t h e s u r v e y . T h e o t h e r g r a p h r e p r e s e n t s a n o v e l s u s p e n s i o n t y p e . T h e r e a r e 8 2 d i s t i n c t s o l u t i o n s

t o G r u b l e r ' s f o r m u l a f o r g r a p h s i n w h i c h t h e w h e e l h u b i s c o n n e c t e d t o t h e c h a s s i s b y t w o c h a i n s

o f l in k s a n d j o i n ts . T h e s e i n c l u d e b o t h p l a n a r a n d s p a t i a l l in k a g e s . R e p r e s e n t a t i v e m e m b e r s o f b o t h

t y p e s a re s h o w n i n F i g . 6 u n d e r t h e c a t e g o r y o f t w o - c h a i n s i n p a ra l le l l in k a g e s . T h e p l a n a r S L A

s u s p e n s i o n , t h e p l a n a r M a c P h e r s o n s t r u t a n d t h e J a g u a r r e a r s u s p e n s io n a r e al l m e m b e r s o f t h i s

f a m i l y . T h e r e a r e 2 2 4 s o l u t i o n s t o G r u b l e r ' s f o r m u l a r e p r e s e n t i n g t h r e e - c h a i n i n - p a ra l le l l i n k a g e s.

A l l o f t h e se l i n k a g e s a r e s p a t i a l a n d s o m e r e p r e s e n t a ti v e m e m b e r s o f t h i s f a m i l y a r e s h o w n i n F i g . 6 .

T h e s p a t i a l S L A a n d t h e s p a t i a l M a c P h e r s o n s t r u t a r e m e m b e r s o f th i s fa m i l y . T h e r e a r e 1 60

d i s t i n c t s o l u t i o n s t o G r u b l e r ' s f o r m u l a r e p r e s e n t i n g g r a p h s o f f o u r - c h a i n i n - p a r a l l e l l i n k a g e s . A s

s h o w n i n F i g . 6 t h e N i s s a n D A R S r e a r s u s p e n s i o n i s a m e m b e r o f t h i s f a m i l y . F i n a l l y t h e re a r e

5 6 s o l u t i o n s t o G r u b l e r ' s f o r m u l a r e p r e s e n t i n g g r a p h s o f f iv e - c h a in i n -p a r a l le l l in k a g e s . T h e

M e r c e d e s f i v e- li n k re a r s u s p e n s i o n b e l o n g s t o t h i s f a m i l y . T h e r e i s n o p o i n t i n l o o k i n g a t l i n k a g e s

w i t h s i x a n d h i g h e r n u m b e r o f c h a i n s b e c a u s e th e s e h a v e z e r o o r n e g a t i v e m o b i l i ty .

I n s u m m a r y , w e h a v e a n a t l a s o f 5 2 4 i n d e p e n d e n t s u s p e n s i o n s g e n e r a t e d b y n u m b e r s y n t h e s i s .

A l l e x i s t i n g i n d e p e n d e n t s u s p e n s i o n l i n k a g e t y p e s l i s t e d i n F i g . 2 a p p e a r i n t h e a t l a s a s n a t u r a l

p r o d u c t s o f t h e e n u m e r a t i o n p r o c e ss . T h e a t l a s a l s o i n c lu d e s s e v er a l n e w s u s p e n s i o n t y p e s . F u r t h e r

d e t a i ls o f t h e e n u m e r a t i o n p r o c e s s m a y b e f o u n d i n [3 ].

4. D I M E N S I O N A L S Y N T H E S I S O F S U S P E N S I O N S

I n t h e p r e s e n t s o l u t i o n w e d i m e n s i o n s o m e o f t h e s u s p e n s i o n s i n t h e a t la s f o r t h e s a m e s e t o f

p e r f o r m a n c e a n d p a c k a g i n g c o n s t r a i n t s . T h e r e s u l t i n g l i n k a g e s a r e t h e n a n a l y s e d i n S e c t i o n 5 t o

s e e h o w w e l l t h e y m e e t t h e s p e c if ie d p e r f o r m a n c e r e q u i r e m e n t s . T h i s p e r m i t s a r a n k i n g o f th e

v a r i o u s c a n d i d a t e l i n k a g e s f o r a g i v e n a p p l i c a t i o n . A n i n d e p e n d e n t s u s p e n s i o n m u s t s a t i s f y

k i n e m a t i c a n d c o m p l i a n c e r e q u i r e m e n t s . T h e k i n e m a t i c r e q u i r e m e n t s t y p i c a l ly s p e c if y c a m b e r ,

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1 1 4 6 M a d h u s u d a n R a g h a v a n

c @ R 0 w c ~ r 0 w

t ra i li n g a m

S t n a l e - c h a i n l i n k a o e s

C R C C R

R R

I R

W R W p

MsePkersml strut p l s u r )

[ = lo n e r h v o - c h a i n i n - o a r a l l e l I i n k a a e s

c @ ~ c ~ : : ~ c ~ ~ ~ cqp :: ~ a

C

R R S R S R

S S C w K ~ S

Jsg--r towr-bsr

S n a t l a l t w o - c h a l n I n - o a r a l l e l I i n k a a e s

R

R W

skort - long arm p O u r )

R - r e v o l u t e j o i n t , W w h e a l h u b

P - p r i s m a t i c j o i n t , C ~ , r t e x ) .

K - H o o k e o l n f , c h a s s is I

S - s p h e r i c a l j o i n t , C e d g e ) - i

r v l i n d r l c J o i n t I

W R W W

P s p

w w

K C K R P

W

W W W W W

S S S S S

s b m 4 oe S a rm S o a f l s l t h r e e - c h a i n I n - o s r a l l e I i n k a a e s Mscl~erson

s t r u t

W - S - W - S - W - - S - - W

S P a t i a l fo u r - c h a i n i n - D a r s I le l I i n k a a e s N i s u n D A R S

K K K K

W W

S S S

8 p a l l a l f i v e - c h a i n i n - P a r a l l e l I i n k a o e s M er ce de s f l ~ - I i nk

F i g 6 A n a t l a s o f i n d e p e n d e n t s u s p e n s i o n s

track, and toe changes of the wheel versus jounce-rebound. Refer to Fig. 7 for a description of

camber, track, and toe. Camber and toe influence the stability of the vehicle during cornering

because they determine the magnitude of the lateral force available at the tire-road contact patch.

Track change during jounce- rebound determines the extent of vehicle roll during cornering. The

suspension (mechanism, spring, shock-absorber) may be treated as a generalized spatial structure

described by the following equation

~ = ~ r , 1 )

where 8~ is a 6 x 1 vector representing the force and moment exerted by the environment (road)

on the suspension at the tire contact patch, 6 ~ is a 6 x 1 vector representing the displacement of

the wheel as a result of 6~~', and ~ is a 6 x 6 stiffness matrix representing the suspension. The

compliance requirements are usually specified by prescribing values for the entries of ~ at the

nominal operating position of the suspension. In this paper, we work with only the kinematic

requirements. Further, we restrict the discussion to front-view kinematics, i.e. camber and track

only. Toe can always be added on to a system designed in this way by including an appropriately

located steer or kingpin axis.

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Dim ension al synthes is of indepe ndent suspension mecha nisms 1147

I

t r a c k

t o p v i e w

vertical

_ i e e l p l a n e

l o n g i t u d i n a l

L .~ j , , ~

axls of vehlcle v ll ne of

I n t e r s e c t i o n o f

w h e e l p l a n e

e n d r o a d s u r f a c e

T r a c k C a m b e r a n d T o e

c a m b e r I n c l in a t i o n o f w h e e l p l a n e t o t h e v e r t ic a l

t r a c k l a t e ra l d i s t a n c e b e t w e e n c e n t e r s o f t ir e c o n t a c t o f a p a i r o f w h e e l s

t o e a n g l e b e t w e e n l o n g i t u d i n a l a x i s o f v e h i c l e a n d li n e o f i n t e r s e c ti o n

o f w h e e l p l a n e a n d r o a d s u r f a c e

Fig. 7 . Cam ber t rack and toe .

T h e w h e e l m o t i o n r e l a t iv e t o t h e c h a ss i s m a y b e d e s c r i b e d a n a l y t i c a l l y b y a f f ix i n g c o o r d i n a t e

s y s t e m s 2~ a n d E t o t h e c h a s s i s a n d t h e w h e e l , r e s p e c t i v e l y . L e t X a n d E b e l o c a t e d s o t h a t t h e y

a r e c o i n c i d e n t a t t h e n o m i n a l w h e e l p o s i t i o n z e r o j o u n c e ) w i t h t h e ir x a x e s a l o n g t h e r o a d s u r f a c e

a n d t h e i r y a x e s n o r m a l t o t h e r o a d s u r f a c e a s s h o w n i n F ig . 8. L e t d x , d y ) a n d 0 d e s c r i b e a g e n e r a l

d i s p l a c e m e n t o f E i n 2L F o r t h e e x a m p l e s i n th i s p a p e r t h e d e s i r e d w h e e l m o t i o n i s s h o w n i n F i g . 9.

I t i s a s t r a i g h t l i n e i n t h e

dy O

p l a n e e n s u r in g z e r o t r a c k c h a n g e a n d a l i n ea r c a m b e r c h a n g e . T h e

v a l u es o f d y a t fu l l j o u n c e a n d f u ll r e b o u n d a r e + 8 0 m m a n d - 9 5 m m . T h e s lo p e o f th e c a m b e r

c u r v e i s 3 / 8 0 / m m . T h e r o o m a v a i l a b l e f o r p a c k a g i n g t h e s u s p e n s i o n i n o u r e x a m p l e i s s h o w n i n

f r o n t v ie w

/ / / 7 E J x

Fig. 8 . Coordinate sys tems.

e

3 d e g .

95

Jounce

x

Fig. 9 . Desi red wheel mot ion.

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1148 MadhusudanRaghavan

7 0 0 m m ~

Fig. 10. R oo m for packaging suspension.

f r o n t v i e w

Fig. 10. Th e fam i ly of l inkages syn thes ized in thi s sec t ion is sho wn in Fig. 11. These l inkages

c ons t i t u t e t he s i mpl e s t one - a nd t wo-c ha i n l i nka ge s i n t he a t l a s . The p re se n t syn t he s i s p rob l e m i s

a n i ns t anc e o f t he c la s s ic a l r ig i d -bod y gu i da nc e p rob l e m . I t is we ll kno wn t ha t t he d i me ns i ona l

syn t he s i s o f a n i n -pa ra l le l l i nka ge fo r r i g i d -b ody g u i da nc e re duc e s t o t he se pa ra t e syn t he s is o f t he

i nd i v i dua l c ha i ns c ompr i s i ng t he l inka ge . As a r e su lt , t he de s ign a nd e va l ua t i on o f t he l inka ge s i n

F i g . 1 1 m a y b e a c c o m p l is h e d b y m e r e l y s y n th e s iz in g t h e f o u r c h ai n s R - R , R - P , P - R , a n d R

c omp r i s ing t he se li nka ge s. Th i s c ha i nwi se de c ou pl i ng i s a lso t rue fo r spa t i a l l inka ge s . The ge ne ra l

t he ory a nd de s i gn e qua t i ons fo r t he va r i ous c ha i ns a re a va i l a b l e i n C ha p t e r 8 o f [4 ] a nd i n [5 ].

4.1. R R chain design equations

The pos i t i on ve c t or o f a n a rb i t r a ry po i n t P i n E a s s e e n in 2 : is

w h e r e

d x )

o y + d y

( c o s 0 - s i n 0 ~

R0 = \ sin 0 co s 0 ]

i s the chan ge oi: bas i s mat r ix re la t ing E an d 2;, and ~ is the p os i t ion vec tor of P in E . I f the R

j o i n t c on ne c t i ng t he R -R c ha i n t o E i s t o be l oc a t e d a t P on E , t he n P mu s t lie on a c ir c le in 2 :

a t each des ign pos i t ion. Analyt ica l ly thi s i s s ta ted as

d x ) _ f t . d x ) _ f i = k 2

( R [ J + k , y , ] } ( R + \ d y } , ]

(2)

whe re subsc r i p t i i nd i c a te s t he i t h de s i gn po s i t ion , ~ i s the p os i t i on v e c t or o f t he c e n t e r o f t he c i rc le ,

a nd k i s t he r a d i us o f t he c ir cl e. Ve c t or s ~ a nd r a re e a c h e qu i va l e n t t o t w o sc a l a r unkn ow ns , k

i s e qu i va l e n t t o one a n d so we ma y de s i gn fo r a m a xi m um of f ive de s i gn pos i ti ons .

c

s w i n g a x l e s h o r t l o n g a m M a c P h e r s o n s t r q t

( p l n , , , ) ( p b e o m r ) N o m e n c l a t u r e

P . p d s m e t lc o l n t

W - w h ~ l h u b

C - c h a s s is

Fig. l l. Plan ar one- and two-chain linkages.

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Dimensional synthesis of independent suspension mechanisms 1149

4 .2 . R - P cha in des ign equa t i ons

F o r t h i s c a se w e m u s t d e t e r m i n e b y t h e d e s i g n p r o c e s s , t h e li ne o r l in e s i n E w h i c h a r e c o n c u r r e n t

i n al l t h e d e s i g n p o s i t io n s . T h e P j o i n t a x i s m a y b e l o c a t e d o n o n e o f th e s e l in e s a n d t h e p o i n t

o f c o n c u r r e n c y i s t h e R j o i n t l o c a t i o n . L e t

U , x + U 2 y

+ U 3 = 0 b e t h e e q u a t i o n o f t h e P j o i n t a x i s

i n E . T h e n i n t h e i t h d e s i g n p o s i t i o n , t h e e q u a t i o n o f t h is l in e i n 2~ i s

(U~ co s 0r - U2 s in

Or)x + (U~

s in 0i + U2

c o s

Oi)y q- d ~,U~ + d~rU2 +

U 3 ) :

0 (3)

w h e r e d x i = - d x r c o s 0 r - d y r s i n 0 r, d Yr = d x r s i n 0 r - d y r c o s 0 r , a n d ( d x r , d y r , 0 r ) d e f i n e t h e

d i s p l a c e m e n t o f E i n Z a t t h e i t h d e s i g n p o s i t i o n . T h e c o o r d i n a t e s o f t h e R j o i n t i n , r , ( XR , y R ) ,

m u s t b y d e f i n i t i o n s a t i s f y ( 3 ) f o r a l l i . T h e r e f o r e ( 3 ) m a y b e r e w r i t t e n a s

U I c o s O r U2

s in

O,)XR + (G

sin 0r + U2

c o s

Or)yR + dx rU l Jr- d~yrU2 +

U 3 =

0 (4)

U ) , U 2, a n d U 3 t o g e t h e r r e p r e s e n t t w o i n d e p e n d e n t s c a la r u n k n o w n s , w h i l e x R a n d y R r e p r e s e n t o n e

e a c h , s o w e h a v e a to t a l o f f o u r u n k n o w n s a n d m a y d e s i g n f o r a m a x i m u m o f f o u r d es i g n p o s it i o n s .

4 .3 . P - R cha in des ign equa t i ons

F o r t h e P - R c h a i n , a t e a c h d e s i g n p o s i t io n , t h e R j o i n t o n E m u s t l ie o n t h e P j o i n t s l in e o f

s l i d e i n Z . L e t

U~x + U2y

+ U 3 = 0 b e t h e e q u a t i o n i n S o f th e P j o i n t s l in e o f s l id e . L e t ( x R , y R )

b e t h e c o o r d i n a t e s i n E o f th e R j o in t . T h e n a t th e i t h d e s ig n p o s i t i o n t h e e q u a t i o n s t a ti n g t h a t

t h e R j o i n t i s o n t h e P j o i n t s l in e o f s li d e is :

U I C O S O r X R

s in OryR + d x r ) + U f f s i n

OrXR + c o s

OryR + dyr) + U3 = 0 (5)

w h e r e ( d x i , d y r, 0 r) d e f i n e t h e d i s p l a c e m e n t o f E i n 2 ;. T h e R j o i n t c o o r d i n a t e s x R a n d y R a r e e a c h

e q u i v a l e n t t o o n e s c a l a r u n k o w n . U ~, U 2, a n d U 3 t o g e t h e r a r e e q u i v a l e n t t o t w o s c a l a r u n k o w n s .

T h e r e f o r e , w e m a y d e s i g n t h e P - R c h a i n f o r a m a x i m u m o f f o u r d e s ig n p o s it io n s .

4.4 . R chain des ign equat ions

T o s y n t h e s iz e t h e R c h a i n s u s p e n s i o n c o n n e c t i n g t h e w h e e l E t o t h e s p r u n g m a s s 2~, w e m u s t

d e t e r m i n e a p o i n t i n E w h i c h o c c u p i e s t h e s a m e p o s i t i o n i n _r f o r a ll t h e d e s i g n p o s i t i o n s . I f ~ i s

t h e p o s i t i o n v e c t o r o f a n a r b i t r a r y p o i n t P i n E , t h e n i ts p o s i t i o n i n S a t t h e i th d e s i g n p o s i t i o n i s

c o s Oi - s i n 0 , ~ . ( d x r ~

s in 0 r c o s 0 , ) P + \ d y J

F o r P t o b e t h e R j o i n t l o c a t i o n , i ts p o s i t i o n i n t h e ( i + 1 ) th d e s i g n p o s i t i o n m u s t c o i n c i d e w i t h

t h a t i n t h e i t h d e s i g n p o s i t i o n . S o

( c o s 0 r + l - - s i n 0 r + ) [ d x r + ) [ c o s O r - s i n 0 i ~ .

[ d x i ~

s in 0r+ ~ c o s0 r +~ P + \ d y r + i = \ s i n 0 r c o s 0 r ) P + \ d y r J

(6)

E q u a t i o n ( 6 ) i s a si n g le v e c t o r e q u a t i o n e q u i v a l e n t t o t w o i n d e p e n d e n t s c a la r e q u a t i o n s . I t

c o n t a i n s o n e v e c t o r u n k o w n ~ e q u i v a l e n t t o t w o s c al a r u n k o w n s . S o t h e R ch a i n m a y b e sy n t h e s i z e d

f o r j u s t t w o d e s i g n p o s i t i o n s .

E a c h o f th e a b o v e c h a i n s w a s s y n t h e s i ze d f o r th e m a x i m u m p e r m i s s i b le n u m b e r o f p re c i s io n

p o s i t io n s . I n a l l c a s e s t h e p r e c i si o n p o s i t io n s w e r e c h o s e n e v e n l y s p a c e d o n t h e w h e e l m o t i o n

r e q u i r e m e n t s o f F ig . 9 . N o n e o f th e c h a in s s y n t h e s i z e d t h u s w e r e p a c k a g e a b l e w i t h i n t h e s p a c e

s h o w n i n F i g . 1 0. S o t h e s y n t h e s is w a s r e p e a t e d w i t h f e w e r d e si g n p o s i t i o n s f o r e a c h c h a i n . F o r

t h e R - R c h a i n t h e f o u r - a n d t h r e e - p o s i t io n s y n t h e s is c a se s w e r e u n s u c c e ss f u l. F o r t h e t w o - p o s i t i o n

p r o b l e m w e h a v e t w o d e s i g n e q u a t i o n s o b t a i n e d f r o m ( 2) w i t h i g o i n g f r o m 1 t o 2 . T h i s i s a s y s t e m

o f t w o e q u a t i o n s i n th e f iv e s c a l ar u n k n o w n s r e p r e s e n t e d b y p , t , a n d k . W e u s e o n e e q u a t i o n t o

e l im i n a t e k f r o m t h e o t h e r e q u a t i o n t o g e t a s in g l e e q u a t i o n i n ~ a n d ] . T h e d e s i g n e r t h e r e f o r e h a s

t h r e e f r e e c h o i c e s s u b j e c t t o t h e p a c k a g i n g c o n s t r a i n t s . T h e p a r a m e t e r p ~ i s s e t e q u a l t o 0 a n d 3q

i s s e t e q u a l t o - 3 5 0 . ( H e r e p j a n d f r e p r e s e n t t h e j t h c o m p o n e n t s o f p a n d ] , r e s p e c t i v e l y .) T h e

MMT 31/8--F

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r e s u l ti n g e q u a t i o n i n p 2 a n d ~ g i v es t h e l o c u s o f d e s ig n s . T h e d e s i g n p o s i t i o n s f o r t h e t w o - p o s i t i o n

s y n t h e s i s a r e s h o w n i n T a b l e 1.

T h e r e su l t i ng e qu a t io n ( l oc us o f v i a b l e de s igns ) in p2 a n d ~ i s:

- -5.62 1 x 10-~t~p2 + 134.8p2 - 17 5 ~- - 1312.5 = 0

(7)

F o r t h e R - P c h a i n , t h e f o u r - a n d t h r e e - p o s i t i o n s y n t h e s i s e f f o r ts w e r e u n s u c c e ss f u l. F o r t h e

t w o - p o s i t i o n c a s e th e d e s i g n e q u a t i o n s a r e g i v e n b y ( 4) w i t h i = 1 , 2 . O n e o f t h e se e q u a t i o n s i s u s e d

to e l im ina t e U3 f r om the o the r t o ge t a s i ng l e e q ua t ion i n xR, YR, U~, a n d Uz. U2 i s e qu a l t o + 1

a n d xR to - 350. T he s e a r e f r e e c ho i c e s a va i l a b l e t o t he de s igne r . T h e U2 va lue i s c ho se n a r b i t r a r i l y ,

t h e XR v a l u e is c h o s e n t o m e e t t h e p a c k a g i n g r e q u i r e m e n t s o f F i g . 1 0. T h e t w o d e s i g n p o s i t i o n s

u s e d f o r t h i s c h a i n a r e t h e s a m e a s t h o s e f o r t h e R - R c h a i n ( s e e T a b l e 1 ) . T h e r e s u l t i n g e q u a t i o n

in U~ a n d yR, r e pr e se n t ing t he l oc u s o f pa c ka g e a b l e so lu t i ons i s:

- - 0 . 0 5 7 U l y R + 2 . 8 1 0 1 0 4 y R 0 . 7 6 0 U i - - 6 7 . 4 1 = 0

8 )

F o r t h e P - R c h a i n t h e f o u r - p o s i t i o n s y n t h e s i s d o e s n o t g i v e p a c k a g e a b l e s o l u t i o n s . T h e d e s i g n

p o s i t i o n s f o r t h e t h r e e - p o s i t i o n s y n t h e s i s c a s e a r e s h o w n i n T a b l e 2 . T h e s y s t e m o f d e s ig n e q u a t i o n s

i s g i v e n b y (5 ) w i t h i = 1 . . . . 3 . T w o o f th e s e e q u a t i o n s a r e u s e d t o e l i m i n a t e U~ a n d U2 f r o m

the r e m a in ing e qu a t ion . T h i s g ive s a s i ng l e e qu a t io n i n XR, yR, a n d U3. T he pa r a m e te r /-/3 i s s e t

t o 1 ( f re e c ho i c e ) a nd t he r e su l t i ng e q ua t ion i n XR a n d yR i s

--5 .56 3 x 10-Sy ~ - 4.173 10-4yR -- 5.563 10-SX~ -- 0.085XR = 0

(9)

F o r t h e R - c h a i n , t w o - p o s i t i o n s y n t h e s i s d o e s n o t g i v e a p a c k a g e a b l e d e s i g n . F o r o n e - p o s i t i o n

s y n t h e s i s t h e d e s i g n p o i n t i s a t t h e c e n t e r o f th e j o u n c e - r e b o u n d r a n g e , i .e .

( d x , d y , 0 ) = ( 0, - 7 . 5 , - 0 . 2 8 1 2 ) . I f t h e w h e el h u b i s m o v e d t o t h i s p o s i t io n a n d t h e n a n y p o i n t

o n t h e w h e e l h u b i s s e le c te d f o r t h e R j o i n t t h e r e s u l t in g s u s p e n s i o n i s g u a r a n t e e d t o s a t i s f y t h e

d e s i g n p o s i t i o n . T h e p o i n t ( - 3 5 0 , 2 5 0) i n 2~ w a s a r b i t r a r i l y s e le c te d f o r t h e R j o i n t l o c a t i o n , a s

i t m e e t s t h e p a c k a g i n g r e q u i r e m e n t s .

I n t h e f o l l o w i n g s e c ti o n w e s h o w h o w t h e d e s i g n e q u a t i o n s d e v e l o p e d s o f a r m a y b e u s e d t o c r e a t e

s u s p e n s i o n l i n k a g e s w i t h s a t i s f a c t o r y p e r f o r m a n c e a n d p a c k a g i n g . F o r e a c h l i n k a g e , w e s el e ct c h a i n

p a r a m e t e r s f r o m t h e a p p r o p r i a t e s o l u t i o n l o c i f o r t h e i n d i v i d u a l c h a i n s . T h i s d e f i n e s t h e l i n k a g e .

A k i n e m a t i c a n a l y s i s i s t h e n p e r f o r m e d t o d e t e r m i n e t h e a c t u a l t r a c k a n d c a m b e r c h a r a c t e r i s t i c s

o f t h e l i n k a g e .

5 . E V A L U A T I O N O F S U S P E N S I O N S

5.1 . R -R , R - R planar short -long arm suspension)

W e s e l e ct tw o p o i n t s s a t i s f y i n g ( 7) a n d a l s o m e e t i n g t h e h e i g h t a n d w i d t h c o n s t r a i n t s o f F ig . 1 0.

F o r t h i s e x a m p l e t h e f o l l o w i n g c h a i n s w e r e s e l ec t ed :

0 . 0 ) ~ = / - 3 5 0 . 0 ]

Up pe r R - R c ha in : P = 581.8 ' \ 440 .0 J

Table 1. R -R chain design positions

dx(mm) dy(mm) 0()

0.0 36.25 1.35

0.00 5 1 . 2 5 1 . 9 2

Table 2. P -R chain design positions

dx(mm ) dy(mm ) 0()

0.0 50.83 1.9

0.00 - 7.5 -- 0.28

0.0 - 65.83 - 2.46

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1151

0 . 0 ) r = - 3 5 0 . 0

L o w e r R - R c h a i n : P = 2 9 5 .5 \ 2 2 0 .0 J

T h e t r a c k a n d c a m b e r c u r v e s f o r t h is s u s p e n s i o n l i n k a g e a r e s h o w n i n F i g . 1 2. W h i l e t h e c a m b e r

c u r v e i s s a t i s f a c t o r y , t h e t r a c k c h a n g e c u r v e s h o w s a n 8 m m d e v i a t i o n f r o m t h e i d e a l a t t h e

f u l l - j o u n c e a n d f u l l - r e b o u n d e d e x t r e m e t i e s .

5 .2 . R - R , R - P p la n ar M a c P h e r s o n s t r u t)

T h e M a c P h e r s o n s t r u t is c o m p r i s e d o f a n R - R c h a i n a n d a n R - P c h a in . W e s e le c t o n e p o i n t

e a c h f r o m t h e s o l u t i o n l o c i o f (7 ) a n d ( 8 ), r e s p e c ti v e l y , k e e p i n g i n m i n d t h e p a c k a g i n g c o n s t r a i n t s

o f F i g . 1 0. T h e c h a i n s s e l e c te d f o r t h i s e x a m p l e w e r e :

R - R c h a i n :

R - P c h a i n :

0 . 0 ) r = - 3 5 0 . 0

P = 295 .5 \ 220 .0 }

(XR, yR) = ( - 350 .0 , 500 .0 )

(U, , U2, U3) = (2 .289, 1 .0 , 303.87)

T h e t r a c k a n d c a m b e r c u r v e s f o r t h i s s u s p e n s i o n a r e s h o w n i n F i g . 1 2 . T h e t r a c k c u r v e s s h o w s

a d e v i a t i o n o f a s m u c h a s 1 5 m m f r o m t h e id e a l . T h e c a m b e r c u r v e t o o s h o w s s i g n i fi c a n t d e v i a t i o n s

f r o m t h e i d e a l , p a r t i c u l a r l y i n j o u n c e .

5 . 3 . R - R , P - R

T h i s l in k a g e is c o m p r i s e d o f a n R - R c h a i n a n d a P - R c h a in . A s b e f o r e , w e s el ec t p o i n t s f r o m

t h e s o l u t i o n l o c i o f (7 ) a n d ( 9) s u b j e c t t o t h e p a c k a g i n g c o n s t r a i n t s o f F i g . 1 0. F o r t h i s p a r t i c u l a r

e x a m p l e , t h e c h a i n s s e l e c t e d w e r e :

P - R c h a i n :

X R , y R ) = - -

189 .55 , 500 .0 )

R - R c h a i n :

U,, U2, U3) = (2 .677, 1 .0 , 7 .499)

o o ~=

= 295 .5 220 .0 J

C A N D I D A T E S

C R

C R

C R

C R

C R

R W

c P

R w

~lOa

g

- 1 0

t m o k c h n g e

r a m ) 1 0

i ) .

requ i rement

I s

~ deg)

P E R F O R M A N C E C H A R A C T E R I S T IC S

F i g . 1 2 . R e su l t s o f d i m e n s i o n a l sy n t h e s is .

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The t r a c k a nd c a mbe r c urve s fo r t h i s suspe ns i on l i nka ge a re shown i n F i g . 12 . The t r a c k c urve

show s a 15-20 mm de v i a t i on f rom t he i de a l a nd t he c a mbe r c urve shows a te nde nc y t o ho ok i n

j ounc e , qu i t e si mi la r t o t ha t o f t he p l a na r M a c Ph e rson s t ru t.

5.4. R-P R- P

The t w o R -P c ha i ns c ompr i s i ng t h i s li nka ge we re se le c t ed f rom t he so l u t i on l oc us o f 8 ) sub j e c t

t o t he pa c ka g i ng c ons t ra i n t s o f F ig . 10.

U p p e r R - P c ha in :

L o w e r R - P c h a i n :

XR, yR)

U,, U2,

xR, yR)

G , U2,

- 3 5 0 . 0 , 5 2 0 . 0 )

U3) = 2 .203 , 1 .0 , 253 .67 )

- 3 5 0 . 0 , 2 5 0 . 0 )

U3) = 4 .467 , 1 .0 , 1318 .54)

The t r a c k a nd c a mbe r c urve s fo r t h i s suspe ns i on l i nka ge a re shown i n F i g . 12 . The t r a c k c urve

show s sma ll < 5 mm ) de v i a t i ons f rom t he ide a l. The c a m be r c urve is i nd i s ti ngu i sha b le f rom t he

ideal .

5.5. R-P P- R

The R -P a nd P-R c ha i ns c ompr i s i ng t h i s l i nka ge a re s e l e c t e d f rom t he so l u t i on l oc i o f 8 ) a nd

9) sub j e c t t o t he usua l pa c ka g i ng c ons t ra i n t s .

R -P c ha in : XR ,yR) = - -35 0.0, 500.0)

U~, U2, U3) = 2.289, 1.0, 303.85)

P - R c h a i n : X R , y R ) = - - 15.618, 150.0)

U~, U2, U3) = 10.084, 1.0, 7.499)

The t r a c k a nd c a mbe r c urve s fo r t h i s suspe ns i on l i nka ge a re shown i n F i g . 12 . The t r a c k shows

a goo d m a t c h < 1 mm de v i a t i on) wi t h t he i dea l . The c a mb e r c urve i s a l so fa i rl y we ll be ha ve d .

5.6. P-R P- R

The t wo P-R c ha i ns c ompr i s i ng t h i s l i nka ge a re s e l e c t e d f rom t he so l u t i on l oc us o f 9 ) sub j e c t

t o t he usua l pa c ka g i ng c ons t ra i n t s .

U p p e r P - R c h a in : X R ,yR) = -- 115.36, 400.0)

U~, U2, U3) = 3.532, 1.0, 7.499)

L o w e r P - R c h a i n : x R , y R ) = - - 7 . 0 6 7 , 100.0)

U ,, U2, U 3 )= 15.211, 1.0, 7.499)

The t r a c k a nd c a mbe r c ha ra c t e r i s t i c s fo r t h i s suspe ns i on l i nka ge a re shown i n F i g . 12 . B ot h

t ra c k a nd c a mbe r a re i nd i s t i ngu i sha b l e f rom t he i de a l , ma ki ng t h i s l i nka ge a ve ry a t t r a c t i ve

c a ndi da t e fo r t h i s e xa mpl e .

F i gure 12 p rov i de s a v i sua l c om pa r i son of the pe r form a nc e c ha ra c t e r is t ic s o f t he va r i ous

l inka ge s . A ra nk i ng of t he li nka ge s in t e rms of how we l l t he y ma t c h t he t r a c k a nd c a m be r

re qu i re me nt s i s shown i n Ta b l e 3 .

T h e f o l l o w i n g o b s e r v a t i o n s i n T a b l e 3 m a y b e m a d e f r o m t h e a b o v e r a n k i n g s :

1 . The l inka ge s wi t h t wo pr i sma t i c j o i n t s m a t c h t he t r a c k a nd c a m be r r e qu i re m e nt s o f t h is

e xa mp l e ve ry we ll . In pa r t ic u l a r , t he P - R , P -R ) l i nka ge i s ou t s t a nd i ng i n bo t h c a te gor i e s .

2 . The t wo-c ha i n l inka ge s wi t h a n e ve n num be r o f p r isma t i c j o i n t s i .e . 0 o r 2 ) f a re be t te r t ha n

t h o s e w i t h a n o d d n u m b e r o f p r i sm a t i c j o i n ts , viz . t h e R - R , P - R ) l in k a g e a n d t h e R - R , R - P )

l inkage.

3. The planar two-cha in l inkages a re super ior to the planar s ingle-cha in l inkage .

H o w e v e r o n e s h o u l d b e a r i n m i n d t h a t t h e r a n k in g s i n T a b le 3 m a y c h a n g e i f t h e w h e e l -m o t i o n

re qu i re me nt s c ha nge .

8/11/2019 27_suspension mechanism+

13/13

Dime nsional synthesis of independen t suspension mechan isms 1153

Table 3 . Ranking of candidates

Track C amber

1 . (P -R , P -R ) 1 . (P -R , P -R ) , (R -P , R -P) , (R -R , R -R )

2 . (R -P , P -R ) 2 . (R -P , P -R ) ,

3 . (R -P , R -P) 3 . (R -R , R -P)

4 . (R -R , R -R ) 4 . (R -R , P -R )

5. (R- R, R-P ) 5 . R

6 . (R -R , P -R )

7. R

6. C L O S U R E

T h i s p a p e r d e s c r i b e s th e s y n t h e s i s o f i n d e p e n d e n t s u s p e n s i o n l i n k a g e s s u b j e c t t o p a c k a g i n g

c o n s t r a i n t s . W e e n u m e r a t e a n a t l a s o f i n - p a r a l l e l s u s p e n s i o n l i n k a g e s u s in g n u m b e r s y n t h e si s . W e

t h e n d i m e n s i o n a s et o f se v e n s im p l e p l a n a r s u s p e n s i o n c a n d i d a t e s f r o m t h is a tl a s t o m e e t m o t i o n

r e q u i r e m e n t s w i t h z e r o t r a c k c h a n g e a n d l i n ea r c a m b e r c h a n g e . T h e m e t h o d o l o g y f o r d i m e n s i o n i n g

t h e l i n k a g e s r e l ie s o n t h e s e l e c t io n o f p r e c i s i o n p o s i t i o n s , t h e f o r m u l a t i o n o f d e s ig n e q u a t i o n s

m o d e l i n g t h e l in k a g e , a n d t h e s o l u t i o n o f th e s e e q u a t i o n s f o r t h e l i n k a g e d i m e n s i o n s . W e u s e f e w e r

t h a n t h e m a x i m u m p e r m i s s ib l e n u m b e r o f d es i g n p o s it i o n s fo r e ac h l i n k a g e in o r d e r t o b r i n g t h e

p a c k a g i n g c o n s t r a i n t s e x p l i c i t l y i n t o t h e p r o b l e m . A c o m p a r i s o n o f t h e s y n t h e s iz e d l i n k a g e s s h o w s

t h a t f o r t h e c o n s t r a i n t s i n o u r e x a m p l e , p l a n a r t w o - c h a i n i n - p a r a l l e l l i n k a g e s w i t h t w o p r i s m a t i c

j o i n t s h a v e s u p e r i o r k i n e m a t i c c h a r a c te r i st i cs . T h e i r d y n a m i c p e r f o r m a n c e u n d e r c o n d i t i o n s o f h ig h

l o a d s , v i b r a t i o n s , a n d j o i n t f r i c ti o n , i s y e t t o b e e v a l u a t e d .

R E F E R E N C E S

I. S. Mola,

Fundamentals of Vehicle Dynamics.

General Motors Institute (1967).

2. L. D obrjan skyj and F. Freudenstein,

Trans. ASM E J. Engng Ind.

89B, 153 (1967).

3. M. Raghavan,

SAE Passengers Car Meeting and Exposition

Nashville, Tennessee (September 16-1 9, 1991), also

appears in

Car Suspension Syst. Vehicle Dyn.

SP-878.

4. O. Bottema and B. Roth,

Theoretical Kinematics.

North-H olland (1979).

5. P. Chen and B. Roth,

Trans. ASME J. Engng Ind.

91B , 209 (1969).

S Y N T H E S E P R N O M B R E S E T D I M E N S I O N S D E M I~ C N I SM E S D E

S U S P E N S I O N S I N D l ~ P E N D A N T E S

So mm aire- -So nt expos6es ici l 6tude et la classification structurelle de suspensions fi roues ind6pendantes.

No us avons cr66 un a tlas des m6canismes de suspensions parall~les au moyen d u ne synth~se par nombres.

Tou tes les suspe nsions/t roues ind6pendantes existantes figurent dans Fatlas comme p rodu its naturels du

proc6d6 d 6num 6ration. No us avons 6stabli le dimensionn emen t de sept suspensions possibles tir6es de

cet atlas pour un ensemble donn6 de niveaux de performance et de volumes disponibles. La m6thode de

dimensionnement retenue exige la s61ection de positions de pr6cision, la formulation d 6qua tions

mod61isant le m6canisme et la r6solution d e ces 6quations p our les dimensions du m6canisme. Nous avons

eu recours fi un nom bre inf6rieur au maximum permissible de positions de pr6cision pour chaque

m6canisme afin d obt enir des concepts po uvan t 6tre incorpor6s d ans la volume disponible. U ne

com paraiso n des m6canismes ainsi synth6tis6s est 6galement pr6sent6e.