Synthesis of guidance mechanisms

M~-A.Mack.~

Pemu~,

VoL29. No. 4, pp. 525--$33,1994 Copyrilht~ 1~4 Elsev~ ScienceLul

Printedin Great Britain.All rijhtsreserved

0094-114X/ON$6.00+0.00

SYNTHESIS OF GUIDANCE MECHANISMS K U R T L U C K and K A R L - H E I N Z M O D L E R Technical Universityof Dresden, 01069 Drexlen, M o ~ s t r a f e 13, Germany (Received I March 1991;in revisedform 30 March 1993;rece~oedfor publication3 June 1993) Almraclt--A classical problem in mechanism-synthesisis the guidance of a point Z along a given path with the coordinates x = x(t), y =. y(t), (t, ~ t ~ tz). An open kinematic chain O M X with two revolute pairs in O and M can solve this problem. The constrained motion of this mechanism can be realized by using: --Two hydraulic-cylindersincluding computer control as inputs; one of them between frame and first link, the other between the two moving links. ---Oneinput and geometricalelementsfor describingthe givenpath exactly.Specialdisks and a flexible band complete the open chain to a constrained band-mechanism. Starting from a given path, it is shown how to synthesizea band mechanism which solves this problem. Some examples demonstrate the practical application of such hand-mechanisms.

I. I N T R O D U C T I O N A classical problem in mechanism synthesis is the guidance of a point Z along a given path with the co-ordinates x ( t ) and y(t), (tl ~ t ~ t2). Guidance mechanisms are used in several machines for forestry, agriculture, transport and civil-engineering. These machines can be used for several purposes. The main point is the guidance o f the shovel or a special tool; e.g. for deepening processes, earth-moving and -dumping on a truck. Such a system in general can be considered as a manipulator. The simplest mechanism-structure for realizing a plane curve is an open-loop kinematic chain with two revolute pairs. This mechanism has two degrees o f freedom; the input-angles are ~o, and ¢P2. By using the Gauss-system o f co-ordinates x and y the algorithm of the coupler-curve k, o f point Z can be written according to Fig. I and Rcf. [I] as follows: Z =Z(t)=x(t)+iy(t);

(t, <~t <~t2)

(I)

Z = m e ~', + z, z 2 : (Z - me",,

m' =

(2)

Z - mek',),

(3)

(Z - z e~, Z - z e~),

(4)

~o, = 2 a r c t a n ?.my + ~ / 4 m 2(x 2 - y 2 ) _ (z2 _ x 2 _ y 2 _ m2)2

( x + m)2 + y Z - - z 2

qh = 2 arctan 2 z y ~ ~ / 4 z 2 ( x a - y2) _ ( m 2 _ x 2 _ y2 _ z2)2 (x -4-z) 2 -I- y2 _ z2

"

(5) (6)

The considered motion o f this mechanism can be realized by using two hydraulic-cylinders as inputs; one o f them between the frame and the first link, the other between the two moving links; or both cylinders are connected with the frame, see Fig. 2. (!) One hydraulic-cylinder is connected with the frame, Fig. 2(a): Eo;

E,=E,((o,);

F,=F,(~o,);

Fz=F,((o,,,z).

I~=(E, -- Eo, E, -- Eo); II=(Fz- F,,F,- F,). 525

(7)

(8)

526

KUIT LuCK and KAitL-HEINZMODLER

y

f--

x

Fig. I. Open loop kinematicchain with two degrees of freedom. (2) Two hydraulic-cylinders are connected with the frame, Fig. 2(b):

Eo;

E,=E,((p,);

F(,

t~ = (E, - ,%, E, - Eo);

(9)

F2=Fz(~o,,(p2).

t~ = (F2 - Fo, F2 - Fo).

(10)

For realizing a special plane curve, the hydraulic cylinders can be controlled by a computer. In the next step the open-loop kinematic chain shall be closed by an additional link (Fig. 3). Now the trajectory of point Z can fulfill the given path in only nine points. For generating the path exactly, we can use a band-mechanism. 2. ANALYSIS OF B A N D - M E C H A N I S M S A general band-mechanism with two disk-profiles and a flexible band between the disks is demonstrated in Fig. 4 [I]. The band must be stressed by a weight (e.g. gravity weight) in point Z or by a spiral-spring in the revolute pair M. The analysis of the band-mechanism is done by using the Gauss-system of co-ordinates x and y, which is connected with the frame. The moving ~-system of co-ordinates is connected with the moving disk and inclined under the angle ~. The algorithms for path, velocity and acceleration can be written in the following form [3]: Z = M + z e ~' e'~ -- m e'~' + z e~' e ' ,

( I I) (12)

= i(om e ~' + i ~ z e ~ e I~

2 -- (i~ - ~b2)m e" + (i~" - ~ 2)z e'~ e'~.

(13)

The instantaneous centre of rotation P can be found by using the well-known equation [4]: P = M + i ] ~ l ~ = m e'¢'(! -

~1~).

(14)

(b)

(a)

Y

Z

• z

Z~

zl

Y

M

z

k

._.2 L

S

; Fig. 2. Open loop kinematicchain with two hydraulic-cylindersas inputs.

x

Synthesis of guidance mechani.m~

Z

527

A

O Fig. 3. Four bar linkage with coupler point Z and trajectory.

For calculating the path, we need the angle 9. In this case we must take into consideration the following relations, from the geometrical point view (/-.band-length): ,4 -- m e k' + a ( ~ ) e ~' e ~

B

i=

f:

(15)

=h + b(/1)e 'p

(16)

,4' = dA/da = [a'(=) + ia(a)] e~"e ~3

(17)

B" = d a / d ~ = [b'(/O + ib(/~)] e 'p.

(18)

[b(q)Z+b'(q)~=/Zdq+(,4 - B , A

[a(~)2 + a'(~)z]t/z d~ +

-

B) I/2.

(19)

The three necessary and sufficient compulsory-conditions can now be formulated: o=[a

- -4, -4'],

(20)

,4, a'],

(21)

o = [a -

0=(,4-B,,4-B)-(l-f:[a(~)'+a'(~)2]'/'d~-f[b(~)'+b'(~)z]':2dq) :. (22)

D

Fig. 4. Band-mechanism with two disk-profiles.

~28

KUltr LUCKand K~u..HJn~z Mooum

They can be developed to the following form: [U,e~]ffiu;

U¢~-

U e - ~ = 2iu

(23)

[V,e~]=v;

[ T e ~ - V e - ~ = 2iv

(24)

[W, e '~] ffi w;

We ~ - W ¢ -'~ = 2iw

(25)

with U ---[h + b{B) e~p- m e~ [a'(~) -/a(~)] e ~" v = a(~)[b'~) + ib(#)] e~-'~

W = 2ia[h+ b(p)e ~#u~a

m e~']e-"

2

".

(26)

v ffi [(b'(fl) + ib([~)) ¢~, h + b(/~) e ~ - m e ~']

w--(l - ff [a(~)2+ a'(~'/2d~ - f: [b(q)2+ b'(rl)2]'/= drl)2 - - a 2 -- (k + b ( ~ ) e *a - m e *~, h + b ( ~ ) e #B - m e ' ) .

Equations (13)-(15) are a linear equation-system with unknown variables e '~ and e -'~. From equations (13) and (14) follows: e '~ = 2 i ( u V - v U ) / ( O V - U F ) ffi:F(~, [J, ~o) e -'~ ffi 2 i ( u P On

(27)

v O ) / ( O V - UF).

(28)

g

2iv I

(29)

W

2iw I

the basis of the solving-conditions 0 ffi

and I ffie '~. e-"

(30)

0 =f(a,/~, ¢ ) ffi 2 i ( u ( ~ ' W - VII/) + v ( W U - W U ) + w(KtV - UP))

(31)

0 = g ( . , / 3 , ~p) = 4 ( u V - v U ) ( u P - v~:) + ( [ I V - UP) 2.

(32)

we get

and

Relations (31) and (32) are real functions, equation (27) however is a complex function. The derivations ~ and ~"are needed for the calculation of velocity and acceleration of the coupler-point Z. The derivation of function (27) by time t leads to: i~Je" ffi~b[dF(~, ~, ~o)/d~]

(33)

(/~" _ ~2)e,S ffi ~ [dF(,,,/~, tp)/dtp ] + ~b2[d2F(~t,/], tp)/dtp 2].

(34)

and

Taking into consideration these equations, a simple order for calculation can be given: I. ~p = #~(t), 0 = dcp/dt, O = d2~p/dt =. 2. The nonlinear equation-system (31) and (32) delivers for each angle ~p the attached angles and #. 3. Calculation of e'~,i# : and (i~}"- ~2) e,~ for each triplet(~,/I,~o) by using equations (27), (33) and (34). 4. Calculation of Z, Z, Z and P according to equations (I I)-(14).

s ~ l ~ m of ~

c

,7

~

529

C

Z

o

o:B,o

A

D

D Fig. 5. Simple band-mechanisms, derived from Fig. 4.

In this way the analysis of a band-mechanism (see Fig. 4) can be done. From Fig. 4 we can get several simplifications, see Fig. 5: (a)

a -- a(=) = const

and

A - - t h e h ' + a e ~"+s), A " = ia

e ~'+~),

b = b(p) -- const ra

B--h+be

(35)

B" = ib e 'p

l = a(), - = ) + b(6 - #) +

(b) a=a(~)=const

(A -

B, ,4 - B ) ':2.

ana" b=0

(c) a=a(~,)--const, a=0

and b=b(p)=const.

3. S Y N T H E S I S OF B A N D - M E C H A N I S M S 3. I. S y n t h e s i s

by iterative analysis

In the following the synthesis by iterative analysis shall be demonstrated. We start from the approached straight-line of a four bar linkage. This straight-line (length = 2r) shall be improved (minimum of deviation) by using a band-mechanism, see Fig. 6. The fixed and the moving disk have cylindrical surfaces. Using the given values r, xo, Yo and p, the task of synthesis shall be

formulated according to equation (35) and Fig. 6: fl(X)

= ( Z ( t -- to) - Xo - iyo,

Z(t = to) - Xo - iyo) = 0

(36)

f2(x) = (e(t = to) - z(t = to), e") = 0

(37)

f~(x) = [2(t = to),e"] = 0

(38)

0

0

Fig. 6. (a) Level.luffingjib crane. (b) Band-mechanism as alternative solution.

530

KUIT Luc~ and l ~ u u . - ~ z Mooum f . ( X ) = ( Z ( t = to) + • e" - Z (t = t, ), e*) = 0

(39)

fs(X)

(40)

=, ( Z ( t ,~ to) - • e " - Z ( t = t2), • ~0) = 0

f6 == ( Z ( t = to) + • e ~ - Z ( t = t, ), Z ( t = to) + r e" - Z ( t = t, )) + ( Z ( t = to) - • e ~' - Z ( t = t2), Z ( t = to) - r e " - Z ( t ,= t2) ) = mill [

(41)

These equations have been created to improve the straight-line to an optimum. The vector X of the variables contains eight parameters X = [to, h, t2, ~, m, a, b, i]T.

(42)

Altogether we have six equations; therefore two parameters can be chosen free, e.g. t t and t2. In this case we get an extreme-value-task with secondary conditions on the basis of the following nonlinear equation system: O=ft(to,

O,m,a,b,i)

0 --fe(t0, O, m, a, b, i) 0 =A(to, ~', m, a, b, l)

(43)

0 =A(to, O, m, a, b, i) 0 =fs (t0, ¢, m, a, b, l) with

A,0 A,o A,0 A,0 A,o f~0 A, A, f~ f.. A, f~, O=

A~ f,- f~ f~ f~ f.~ f~ f~ f~ f~ At .f~ At f~ fu

f~ f~ f~

(44)

The two-dimensional multiplicity of solutions gives the possibility to consider secondary conditions, e.g. in the form of un-equations, to get optimal results. 3.2. E x a c t - s y n t h e s i s

of band-mechanics

In the following the exact-synthesis of band-mechanisms shall be demonstrated, see Fig. 7. The geometrical profile of the moving disk must be calculated in such a way, that the given path can be generated exactly. In the first step the main parameters m,z,h,b,l,~b

can be chosen. Starting from equation (I I) the moving parameter ~ can be calculated by e~ = e-~[x(t) + i y ( t ) - m e~*]/:.

(45)

For describing the geometrical profile the following conditions have to be respected: A =me'+a(a)e

~'÷~,

B=h

A ' + [a'(ot) + i a ( ~ ) ] e ~.÷3~,

+ b e ~p,

B" = ib e 'n

i -- I ' [a(~)2 + a'(~)2]"2 d~ + b(6 - / ] ) + (A - B, A - B) '/2. A

(46)

Syathem of t~hm~ ~

53i

D Fig. 7. Band.mechanism with fixed cylindrical disk-profile.

The unknown parameters a, a, a' are determined by the following nonlinear equation system:

O=[a-A, 0 = [B- A,AI Sl

1

(f:

0=(A-e,A-e)-

t-

[a(,~)~+a'(O~l'/~d,~-

(47)

I; [b(r/)'+b'(,/)O'r~de )'

A practical application is shown in Fig. 8. The radius of the fixed disk is b = 0, the movement in vertical direction is carded out by an hydraulic cylinder. Regarding Fig. 7 we can change both disks and the fixed cam-profile must be calculated by using the following equations (see Fig. 9): A = m e ~ ' + a e ~'+s) B = h + b ( p ) e 'n ,4" = ia e *° + a)

i=a(y - e ) +

B" = [b '(fl) + ib (fl)]

I;

/

e"e

[b(~)2 + b'(~)2]'n d~ +(A - B , A - B ) '/~

Fill. S. Buildins-machine with a band-mechanism.

I .

(48)

532

Kuxr Lucz and I O d u . - ~ Mootnt

D

Fig. 9. Band-mechanismwith a movingcylindricaldisk-profile. A further simplification leads to the band-mechanism in Fig. 10. The moving disk has been reduced to a point A. The calculation of the fixed disk is running by the following equations:

A = m d ~ + a e '~, B = h + b ( p ) e

'p

A' = 0 B" = [b'(8) + ib(,8)] ¢'a

1 l

(49)

! = 16 [b01) 2 + b'01)2] I/~d~/+ (A - B, a - B) ~/= Jp 4. F I N A L

REMARKS

Mainly in this paper the analysis and synthesis of band-mechanisms with one degree of freedom is demonstrated. Starting from the task of generating a general path, it is shown how to synthesize a band-mechanism, which solves this problem. The synthesis is carried out in two ways, firstlyby using the iterativ¢analysis and secondly by using the exact method. An example demonstrates the firstmethod.

z

Y

D Fig. I0. A simpleband.mechanismwith one fixeddisk-profile.

Syntbem of t,uidam:e m...-h,,;,,,o

REFERENCES 1. K. Hain, ~ n m ~ e C~tr~ebeleb~. VDI-Verlag D~sJeldorf (1961). 2. G. Geite, W'm. Z. TH Dresden 9, S.597--603 (1959/60). 3. K. Lack and K.-H. Modler, Proc. $YROM "8$, Voi. I-1, pp. 223-230, Backarest (1985). 4. K. Lack and K.-H. Modler, Getr~betechnik.Analy~, Synth¢~, Optmvuerung. Springer, Wien (1990).

SYNTHESE VON FUEHRUNGSMEEHANISMEN Zmammmfamms--Ein klassischesProblem der Getriebesynth~ ist die F~hrung eines Punktes Z auf einer 8esebenen Kurve mit den Koordinaten x - x(t), y - y(t), (tI ~ t ~ t2). Mit einer ofl'enenkinematischen Kette OMZ mit zwei Drehgelenken in O trod M liBt sich diese Problem 16sen. Die zwangJiultge kwegung des Mechanismus kann in folgender Weisse i~.alisiertwerden: --Antrieb dutch zwei gesteuerte Hydraulikzylinder,yon denen einer im Gestcil gelagert ist und der andere zwischen den beiden bewegten Gliedern wirkt. --Fin Antriebsglied und geometrische Elementezur 8enauen lkschreibung der Kurvenffihrtmg. Fine spezielle Kurvenscheibe und ein flexibles Band erginzen die offene kinematighc Kette zu einem Bandgetriebe.Aesgehend yon einer vorgesebenen Kurve wird die Synthese eines einfachen Bandgetriebes aufge~igt. An Hand einiger lkispiele wird die praktische Anwendung solcher Bandmechanismendemonstriert.

533