98ye0196 curvature mechanism

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VOI

41 NO.

2

SCIENCE IN CHINA (Series E

~ p r i l 998

A

unified curvature theory in kinematic

geometry of mechanism

WANG Delun (E &), IU Jian 311

)

and

XIAO

Dazhun

Department of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China)

Received August 18, 1997

Abstract

Based on the differential geometry, the expressions of centrodes and axodes as well as their invariants are

derived. And then, the kinematic meanings of them are revealed. Meanwhile, the properties of a point trajectory in

planar, spherical and spatial motion or a line trajectory in spatial motion are discussed. A main clue for the unified cur-

vature theory in kinematic geometry of mechanism is set up in the form and content from plane to space motion.

Keywords:

cuwature theory kinematic geometry

of

mechanism differential geometry.

The curvature theory in two- and three-dimensional motion has been studied extensively over

the past decades. It is abundant and systematical for planar motion. The curvature theory in space

motion is not expressible in terms comparable to those for the plane yet[ 1. However, many inves-

tigators have made efforts to extend the curvature theory in plane to that in space by means of

many approaches[ - 21, but more light needs to be shed.

The curvature theory in planar motion shows the properties of a point path based on the cen-

trodes and their invariants, but the curvature theory in space motion not only shows the properties

of a point trajectory but also reveals that of a line trajectory. Therefore, we might also expect the

axodes to occupy a similar important fundamental place in spatial instantaneous kinematic geome-

try of mechanism. It is a key problem whether the axodes and their invariants could be used to

describe the properties of a point trajectory and a line trajectory or not. Authors have introduced

the new adjoint approach[15-171 into the curvature analysis of space motion by means of the ax-

odes. Therefore, the purpose of this paper is to lay a goundwork or set up a main clue for the uni-

fied curvature theoretical system of two and three dimensional motion.

1

The cuwature

theory in

planar

motion

The curvature theory in planar motion has been extensively studied by many approaches over

the past decades. This section only shows the systematicness and completeness of the unified cur-

vature theory by the adjoint approach.

Assume that the moving centrode n, and fixed centrode xf for a moving body are known

curves. Their vector equations are

~ , : r , = r , ( s ) , ~ f : r f = r f ( s ) , (1)

where s is the arc length of n, or n f . The moving Frenet frames, r elm) , elmr and 1 r f ,

el f) , ei f) are established at nmand

y

respectively. They are coincident with each other at the

velocity centre

P,

shown in fig. 1. Then, the vector equation of the path F traced by a point of

Project supported

by

the National Natural Science Foundationof China Grant No.

59305033 .



No 2 UNIFIED CURVATURE THEORY IN KINEMATIC GEOMETRY OF MECHANISM

197

a moving body can be written as

FA:RA= r f

+

xle i f )

+

x2eif )

=

r f

+

r(cos0e$')

+

sin0ehf)>,

2)

where

(

x

1

x 2 is the coordinate of point in the mov-

ing frame { r f , elf) ,eif) ( 6 is the polar coordinate of

point A , which are all the function of arc length s of x i

Differentiating eq. ( 1 with respect to s and simplifying

it by the fixed point

condition[181,we have

R;

= k

( -

x2elf)

+

x le i f ) ) , ( 3 )

~ R A

where R i

=

s

k

is defined as the induced relative

curvature, and

4 )

Fig. 1

k

*

=

k f

k .

In the above equation, k f and k, are the relative curvature of x f and n respectively, and they are

both invariants in kinematics. The velocity expression of point A of the moving body can be writ-

ten as follows by the velocity pole P :

VA = w x r

=

wk x ( x l e j f )+ z2e i f ) )= w(- x2ejf)+ x le i f ) ) ,

where w = wk, which is the angular velocity vector of the moving body, and w is the angular ve-

locity, k = e(lf)

x

eif) , r is the vector distance from the velocity pole P to point

A

Meanwhile,

another expression of VA can be derived by eq. (3 ) , which is

Comparing the above two expressions of VA, we obtain the relationship between

k

and w :

w = k

&

d t '

which reveals the kinematic meaning of k *

.

Differentiating eq. 3 ) with respect to s again, sim-

plifying it and substituting it into the curvature equation of a c ~ rve ~ ' ,we can give the relative

curvature equation of point path rAs

If the curvature k is replaced by the curvature radium p another expression of the above equation

is

,o(r + Dsin0)

=

r 2 ,

(6b)

where

= 1 /k

is defined as the diameter of the inflection circle. The above two equations are

the famous Euler-Savary equations, whose geometrical meaning is shown by a circle, point and

OA

in fig. 1 Differentiating eq. (6a) with respect to s once and twice, we can get the expres-

sions of k and k . The equation of cubic stationary curve can be given as follows by letting k'

= o

1

1

-

Msin0 Ncos0'

7)

where



198

SCIENCE IN CHINA Series

E ) Vol.

41

The Burmester's points in the moving body meet

k ' =

0 and

k =

0, which are corresponding to

1 1

-

MsinB NcosB'

9 )

tg4e

+

cltg3e

+

~ ~ t ~ ~ e~ ~ t ~ e

C

=

0,

= -

where c l = ( l - k m M ) N / M , c 2 = ( 1 + N') , c s= ( M ' N + ~ M ) N / M ~ ,

=

( k m M - 2 ) -

N ~ /

~ .

owever, there are less than four Burmester's points in the moving body at an instant

for planar motion, or even no oner1*]

2

he

invariants of axodes

The spatial motion of a rigid body can be visualized as the body slides along and rolls around

the instant screw axis (ISA)

.

Meanwhile, ISA generates two ruled surfaces Z f in the fixed frame

o

i f j f f and

2

in the moving frame om Jmkm of the moving body, which are called the

fixed axode

Zlf

and the moving axode

8

respectively. Provided that

Z f

and

Z,

are known sur-

faces, suppose that their vector equations are

Zm:

R m

= rm+ ,US,; Zf:

R f =

rf + ,usf,

(10)

where r, and

rf

are the vectors of striction curves of Z and Zf respectively, S and Sfare unit

vectors of ISA in of

f j f

f and om Jm km, or generators of

2

and

Sf p

is a parameter of

2

and

Zf.

The Frenet frame r f , e lm ), eim) , eim) of

2

is set up by

where a, is the arc length of the spherical image curve of S . In the same way, the frame

{

r f ,

elf) ,ear),e$') of Z f can be estabished. These two frames are coincident with each other at an in-

stant. That is d a f =

do

denoted as do hereinafter. Then, the vector equation of the trajectory

PAtraced by point A of the moving body can be expressed as

rA:A

= rf +

xle i f )

+

z2eIf)+ x 3 e('),

(12)

where XI,~ 2 ,3 is the coordinate of point

A

in the frame

r f

elf) , eif) ,eSf) (see fig. 2

)

.

Differentiating eq. (12) with respect to a and simplifying it by the fixed point

condition^ ^',

we

obtain the tangent vector of the trajectory

FA:

I?

a elf)

+

p *

-

s3eLf)

+

x2e4 ),

(13)

where Ri = d R A / d a , and

a = a f a,, p

=

Pr

P m -

(14)

a * and

/?

are defined as the induced construction parameters, and a,, P Y, a f, /If,

Y f

are

the construction parameters of 8 and Z f respectively. They are all invariant^ ^'. Each of them

has a special geometrical meaning.

In order to show the kinematic meaning of

a

and

,

we derive two kinds of velocity ex-

pressions for a point of the moving body.

VA= V + a , X r

=

zrelf)

+

w - x3eIf)

+

x2e l f ) ) ,



No.

2 UNIFIED CURVATURE THEORY IN KINEMATIC GEOMETRY O F MECHANISM 199

where

V =

velf) and =

weif ,

which are the

angular velocity, and translation velocity of the

moving body when

2

rolls about and slides along

ISA, is the vector distance from the striction

point of Bf to point A or = x le l f )+ .z2eif)+

x if) Comparing the above two expressions, we

have the following relationship among w

v

and

k m

a , P * :

v = a = ,I3

(15)

d t d t

'

which are corresponding to eq. 5

)

in planar mo-

tion, and show the kinematic meaning of the in-

duced construction parameters a and

P *

3

The curvature theory of a point trajectory

For the convenience in discussing the proper-

ties of a point trajectory

PA

in space motion,

he

Frenet frame RA, EA ,E2, E3 is set up for the

Fig

2 .

trajectory rAs follows:

According to the above equation, we have the normal curvature

n

and geodesic curvature kg of

FA

by differential

:

where

x2

= rcosO, cosOg = ( a * + 8 Y)/D,,

D,

= [ ~ ~ z :( a *

+ P *

~ ) ~ . ] l ~ / p * ~ .

If the geodesic curvature kg is replaced by the geodesic curvature radium p the second expression

of eq. (17) can be rewritten as:

kg = [ ( r - Dgcos(O

+

Og)I/[r2

+

( a * / / ? *

1 1.

(18a)

The geodesic curvature kg is replaced by pg in the above equation, which leads to

p g [ r

-

Dgcos(B + O,)] = r Z

+

( a * / p *

1 .

(18b)

The form of the above equation is similar to that of eq. (6b ) , which is called a geodesic Euler-

Savary analogue. There is a family of circles in the moving body if kg= 0, whose equation is

= Dgcos(B+ B,), which is defined as the geodesic inflection circles, and

D,

is its diameter and

varies with X I , shown in fig.

3 .

Since k2= k: + k i for the curvature k of F and el f) perpen-

dicular to

E3,

we have = kg if and only if k ,=O. In this case, the first expression of eq. (17)

is zero, and the second one can be rewritten as follows:



200 SCIENCE IN CHINA (Series E Vol. 4

i

[ + ( a P ) r2sin6 a DBsin(6 O , ] = 0

k = [ ( r D,cos(O 8,)] /[rZ ( a * / ~ 2] . (19)

This equation is called Euler-Savary analogue of a point

trajectory in spatial motion, shown in fig. 3. It should

be noticed that any point trajectory may not meet the

above Euler-Savary analogue until the point is on the

surface determined by k,

= 0

in the moving body at the

instant. In other words, there exists a geodesic Euler-

Savary analogue for any point trajectory in spatial mo-

tion, but Euler-Savary analogue only for some point

trajectories.

Fig.

3 .

In particular, the spatial motion of a rigid body is

a spherical motion if the axodes are cones

(

a f= a m= 0, yf= Y, = 0 ) . Then, a point trajectory

FA s a spherical curve.

In order to show its special properties, we redefine the moving frame

~ R A , l , E z , E d of as

E l

= ( -

x3e i f ) z2e i f ) ) / r ,

E2

= ( xl el f) x2eif) z 3 e i f ) ) / ~ ,

j= El X E 2 .

(20)

The curvature of rAs k

=

(k: ki) 2. The cubic stationary curvature in spherical motion will

be revealed by k = k , = 0, which takes the form of

1

1

ctgF = Msine NcosB

(21)

The above expressions are all similar to those in planar motion respectively. Furthermore, the

points of the moving body, which trace trajectories with k = O and kt = 0, can be determined by

eq.

(

17) and its derivative with respect to a . The result is

2 PmM M'N

3Mtg0

(1 t$0)[ M2

1 N't$O P m M

1

M ~ N N2 MN

tg30

Since the equation is a sextic algebraic equation, there are at most six real roots, or there are at

most six Burrnester's points in the moving body for spheric motion at the instant.

4 The curvature theory

of line

traectory

A line trajectory

l

raced by a line

L

of the moving body which passes through point A can

be considered as an adjoint ruled surface of the fixed axode Bf (see fig. 2), and its vector equa-

tion can be given as

where RA is the vector from of to point A, its expression is eq. (1 2) ,

1

is the unit vector of line

L and its components in the frame r f , eif) ,eif ), eif) are

(

Z 12, 13), or 1; Z

+

1; = 1

u

is

a parameter of the ruled surfaces Zl Differentiating with respect to a and simplifying it by the



No

2

UNIFIED CUR VATUR E THEORY IN KINEMATIC GEOMETRY OF MECHANISM

201

fixed line conditions[16', we have

1' P *

-

z 3 e i f ) ~ ~ e 4 ' ) ) .

Then, the striction vector of a line trajectory Zl can be written as

rl

R A bll, bl

R; 1 / l f 2 ,

where bl is the distance between the striction curve and the directrix of Z1 along the generator L ,

or striction directrix distance for short. For convenience in researching the properties of a line tra-

jectory Zl, the Frenet frame rl. E{'), E$ , E$') is established at the ruled surface Zl as

~ { l 1,

~ 4 ' )

1'1 I 1 1 ~ 4 ~ { l ) ~ 4 ' ) .

(27)

Since E ; ) is the normal n of Z1 at the striction point[161, n is perpendicular to eIf) according to

eq. (2 5), that is, the common normal between

L

and ISA is just n , or the perpendicular foot on

L

is the striction of Z1 and that on ISA is point B . We designate the distance between two per-

pendicular feet as h , and the distance from the striction point

P

of Bf to point

B

on ISA as

p ,

that is,

Substituting the above equations into the Frenet formula of differential

we obtain

the three construction parameters:

[ Z ~ p* ( 1 ~ : ) l / ~ h b ; ] / [ ~ * 1 1:)1/2],

pl

[13

+ p*

( 1 1:> 1~I/ [p* 1 11)3/2],

(29)

Yz

a* /p* l lh / ( l 1 : )1 /2 .

The above three construction parameters are all invariants. They determine the properties of the

line trajectory completely. However, q

pl

and yl will satisfy the following equation if Zl con-

tacts with a fixed axis constraint ruled surface

2,

corresponding to a binary link n the sec-

ond orderLg1.

This equation is defined as the Euler-Savary analogue of a line trajectory in spatial kinematics. Its

geometrical meaning is shown in fig.

4.

The first expression of eq. (30) reveals the relationship

between the direction of line

L

of the moving

body and the direction of the axis of Z,,, and

the second one describes the distance from the

line L to the axis of 8 . Since PC is a con-

stant, the line trajectory B1 must have p l 0 if

Z and Zcc are in the third order contact. By

eq . 29 )andle t t ing l l=cos~ ,

1 2 = s i n ~ c o s 0 ,

s

we have

1

1

-

ctgy, Msine

Ncos6'

(31)

This equation is the same as eq. (21) and shows

Fig



202 SCIENCE IN CHINA (Series E) Vol. 41

that there is a cubic stationary curvature curve for the spherical image of the line s unit vector of

the moving body according to the geometrical meaning of

PL

Furthermore, the line trajectory

Z

contacts with the osculating fixed axis constraint ruled surface Z in the fourth order, which leads

to

Pt

=0, and

p l=

0, or eq.

23)

appears again. I t means that there exist at most six line s di-

rections in the moving body at the instant and the line trajectory contacts with

2

in the fourth

order if r o = 0 and

rJJo

0, or a L -

=

0 and

afJl

-

=

0. These lines are called the

Burmester s lines in spatial kinematics.

5 onclusion

The Euler-Savary equation of planar motion can be extended to a geodesic Euler-Savary ana-

logue for a point trajectory and the Euler-Savary analogue for some point trajectories at an instant

in space motion, and a line trajectory in spatial motion has similar properties to a point trajectory

in spherical and planar motion, such as Euler-Savary analogue, cubic curve of the stationary cur-

vature and Burmester s direction or Burmester s line.

References

Hunt, K . H .

Kinematic Geometry of Mechanisms

Oxford: Clarendon Press, 1978.

Bai, S. X . Advanced Mechanism (in Chinese), Shanghai: Shanghai Science and Technology Press, 1981.

Roth, B., Finite-position theory applied to mechanism synthesis, A S M E 1967, 89E: 591.

Chen, P . Roth, B., A unified theory for the finitely and infinitesimally separated position problems of kinematic synthesis,

A S M E 1969, 91B: 203.

Tsai,

L.

W., Roth, B., Design of dyads with helical, cylindrical, spherical, revolute and prismatic joints,

Mechanism and

Machine Theory 1972, 7( 1) 85.

Chiang, C. H., Kinematics of Spherical Mechanisms Cambridge: Cambridge University Press, 1988.

Yang,

A.

T., Roth,

B.

Higher order path curvature in spherical kinematics, A S M E 1973, 95B: 612.

Kirson, Y . Yang, A. T. Instantaneous invariants in three-dimensional kinematics, A S M E Journal of Ap plied Mechan-

ics 1978, 45: 409.

Veldkamp,

G.

R. Canonical systems and instantaneous Invariants in spatial kinematics, Journal of Mechanisms 1967, 3:

329.

Veldkamp, G. R. ,

n

the use of dual numbers, vectors and matrices in instantaneous spatial kinematics,

Mechanism and

Machine Theory

1976,

:

141.

McCarthy,

J .

M., Roth, B . Instantaneous properties of trajectories generated by planar, spherical and spatial rigid body

motion,

ASME Journal

of

Mechanical Design

1982, 39.

Yang, A . T. irson, Y . , Roth, B., On a kinematic curvature theory for ruled surfaces,

Proceedings of 4t h Wm

Congress on Mechanism and Machine Theory New Castle: Upon Type, 1975, 737-742.

Sasaki, S. Differential Geometry (in Japanese), Tokyo: Kyolitsu Press, 1956.

Wu, D. R . Differential Geometry (in Chinese), Beijing: Higher Education Press, 1981.

Wang, D.

L.

Liu,

J .

Xiao,

D. Z.

Kinematic differential geometry of a rigid body in spatial motion-I, Mechanism and

Machine Theory 1997, 32(4) 419.

Wang,

D.

., Liu, J . Xiao, D .

Z.

Kinematic differential geometry of a rigid body in spatial motion-11,

Mechanism and

Machine Theory 1997, 32(4 ) 433.

Wang, D. I-. Liu,

J .

Xiao, D.

Z.

Kinematic differential geometry of a rigid body in spatial motion-111, Mechanism and

Machine

Theory,

1997, 32(4 ) 445.

Wang, D. L . , Xiao, D. Z . Liu, J . . Ball s curve and burrnester s curve for four-bar linkage, Journal of Dalian Unive rsity

of Technology (in Chinese), 1994, 34(4 ): 411.

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