An application of graph theory and nonlinear programming to the kinematic synthesis of mechanisms

Mech. Mac& Theory Vol. 26. No. 6. pp. 553-563. 1991 Printed in Great Britmn. All rights reserved

0094-114X,r91 $3.00 ÷0.00 Copyright ~ 1991 Pergamon Press pk:

AN APPLICATION OF GRAPH THEORY AND NONLINEAR PROGRAMMING TO THE KINEMATIC SYNTHESIS OF MECHANISMS D A M I R V U C I N A t and F E R D I N A N D F R E U D E N S T E I N Department of Mechanical Engineering, Columbia University, New York, NY 10027, U.S.A. (Receired May 1989: receivedfor publication 9 January 1991)

Almract--In this investiption a systematic"start-from-zero" approach for the design of mechanisms has been described and applied to the creation of an automotive windshield-wiper mechanism. Following

specification of the design goals and restrictions, the potentially useful kinematic structures have been generated with the aid of graph theory. A preliminaryfunctional screeningof these structures has yielded several potentially useful mechanisms. The type synthesis was followed by dimensional synthesis, formulated as a nonlinear programming problem. Physicaldesign goals and constraints were transformed into the objective and penalty functions and then optimized. The final result was a proportioned potentially optimal mechanism. An alternative approach to the dimensional synthesis of the generated kinematic structures in terms of precision points has been discussed as well.

I. I N T R O D U C T I O N Kinematic synthesis may be defined simply as designing mechanisms which perform desired kinematic and dynamic tasks, i.e. transferring motion from source to output in a prescribed manner. The design may be subdivided into two parts: type synthesis (which determines kinematic structure) and dimensional synthesis (which determines proportions). It is evident that these two aspects o f the design process need to be coordinated. A direct approach is not feasible: the structure needs to be determined before the dimensional synthesis can take place, while any structure cannot usually be examined and evaluated for functional acceptability before the corresponding optimal dimensions have been determined. The approach to type synthesis followed here is the concept o f "creative design according to the separation o f kinematic structure and function" [I] which utilizes graph theory for structural representation. This concept has been applied in conjunction with numerical optimization for dimensional synthesis and optimization. The particular goal in this investigation will be the design o f a single-blade automotive windshield-wiper mechanism with optimal sweep and wiping action, inspired by an anonymous sketch [2].

2. C O N C E P T O F C R E A T I V E D E S I G N OF M E C H A N I S M S A C C O R D I N G T O T H E S E P A R A T I O N OF T H E K I N E M A T I C S T R U C T U R E AND F U N C T I O N . GRAPH THEORY FUNDAMENTALS The basic idea of this approach is to consider kinematic structure and function as though independent in the initial design stage. Kinematic structures can then be generated automatically and in many ways independently o f their functional characteristics. The corresponding functional properties are evaluated at a later stage. If graph representation o f kinematic structures is used, the following essential steps are involved: • Generation of all graphs up to the desired degree o f complexity. • Labelling the edges o f each graph in all nonisomorphic ways. • Assigning ground, input and output link locations. tPermanent address: University of Split, Yugoslavia. 553

554

DAMIR VL'c1Na and FERDINAND FIIEUDENSl~N

(a)

(b) 5

s

•

"< 4

d

J Fig. I. (a) A labelled graph with two independent loops (I. It) and peripheral loop (111). (b) Two trees of the graph shown in Fig. l(a).

A graph can be loosely defined as a set of vertices, some of which are connected by edges. Hence they can be well-suited for representing the interdependence between elements of structures and abstract groups. Some basic concepts and definitions are as follows: • Vertices may be connected by edges. An edge is said to be incident at a vertex if the vertex is the endpoint of that edge. • The degree o f a vertex is the number of edges incident at that vertex. • A path between two vertices is a sequence of edges connecting the vertices. • A circuit (cycle) is a closed path, in which each vertex is traversed exactly once. Two or more graphs are isomorphic if there is a one-to-one correspondence between their edges and vertices which preserves incidence. • A tree is a subgraph of a connected graph containing all of its vertices and no circuits. • An independent loop (loop has the meaning of the concept circuit here) may be defined as one that does not circumscribe any other loop (for planar graphs)

The basic equations of graph theory can be obtained as follows: For any graph, let e= v -L, = v, =

number number number number

of of of of

edges, vertices, independent loops, vertices of degree i.

From Fig. ! we have e , , = v - !, L, = e - v + 1,

2 . e = Y. i .v,.

(I) (2)

(3)

Let the mechanism be represented by a graph as follows: The graph of a mechanism with binary joints is a graph in which the vertices correspond to links, edges correspond to joints, and the edge connection between vertices corresponds to the joint connection of links. In terms of the kinematic structure equations (!)-(3) and the basic mechanisms" degree-offreedom equation become: L,,d = J -- I + 1,

(4)

2 -j = ~ i- t,,

(5)

F = Z f - A. Li~,

(6)

Kinematic synthesis of m~hanisms

555

where j--number of joints in the mechanism, /--number of links, /,--number of links with i joints, f--degree of freedom of ith joint. F--degree of freedom of mechanism. The number of independent loops (L,~) of a graph may be regarded as an appropriate measure of the complexity of the corresponding mechanism.

3. TYPE S Y N T H E S I S

OF A W I N D S H I E L D - W I P E R

MECHANISM

Several restrictions and assumptions on the kinematic structure of mechanisms are obeyed in this design.

Structural and functional specifications (I) Only single-degree-of-freedom, plane, closed-loop mechanisms are considered. (2) Only revolute, prismatic and gear pairs may be used. (3) The maximum number of prismatic and gear pairs in the mechanism is limited to two and one, respectively. (4) Structures capable of motion due only to special dimensions will not be considered. (5) No single link can have more than one sliding joint (P). f6) The motion of the link attached to the wiper blade has to be symmetric in the regions of angular displacements from [0, n/2] and In/2, n] (Fig. 9). (7) The complexity of the structure of potential mechanisms will be limited to L,~ = 2. {8) A single wiper-blade serves the entire windshield, which can have a range of proportions. The above specification yields the following restrictions on the kinematic structure of the automotive windshield-wiper mechanism:

(,4) If only R, P pairs were arailable. From equations (4) and (6) with f,=!

or

J=

Ef,=J:

31 - 4 2

(7)

(B) If one G-pair is included. Similarly, with

~

---j -I- l,

we have 3•-3

J =

(8)

4

Generation of kinematic structures: preliminary selection Given the above specification and with the aid of systematic graph generation, the structures of Figs 2-8 which potentially satisfy the design criteria have been generated (preliminary screening).

556

DA~uR VUCINaand FF.RDINA.NDFREUDE.~STEIN

O13~

Graph structure 2

R

1

R 4 Fig. 2(al

ECH

Y

I

e=OA h-CO d-BC t.DE p=BE

3

Xc

Fig. 2(b) MECH 2 Y E

Graph structure

A

1,,EB

2 R3

1

R

4

Fig. 3(al

Fig. 3(h) MECH 3

~" C 5 Graph structure

d=OA f=BC p=CE I,CD e-OE

3 2

4

P

R

Fig. 4~a)

Fig. 4(h) MECH 4 Y d=OA f=AC

D

e=OB ~

1

Graph structure 3

2 R

4 R P

In B there is no joint between links I and 4 Fig. 5(a)

c@

' Fig. 51b)

Kinematic synthe~ of mechanisms

557 MECH 5 s

d=OA f-AC e=BC

';"&

Graph structure 0

3

I¢

2 R

X

4 p R

Fig. 6(a)

Fig. ~b)

MECH 6

Graph structure 5

~x

d-O

R I ~ f R 5R

p-BE

Fig. 7(a)

Fig. 7(b) MECH 7

I=AC hOB I=EO

Graph structure

RI~-JP5R

~ ~

=

P

C

In B there Is no J o i n t between links 4 and 5 Fig. 8(a)

~

~

~

l

~

~

4

Fig. 8(b)

Figs 2-8. Graphs and corresponding mechanisms for candidate kinematic structures I-7. 4. N U M E R I C A L O P T I M I Z A T I O N IN C O N S T R A I N E D R E G I O N S T H E E X T E R N A L PENALTY-FUNCTION CONCEPT In the penalty-function approach to design optimization, the goal is to find the position ( x . ) and the value f ( x . ) of the minimum of a function f(x), MMT 2~f~-B

x (~ R',

(9a)

DAMIR VUONA and FERDZNANDFaBUDENS'na,~

558

in the n-dimensional space of real-numbers. Since the function to be optimized will be created from and related to such abstract concepts as "'design quality", "technical peroformance", etc. [see equation (!1)], analytical optimization methods based on Aft L..., = O.

(9b)

are not feasible. In addition, a fairly large number of variables as well as a rather complex function [equation (9a)] can be expected. The total goal function ("objective function"), f(x), is composed of the partial goal functions as follows: L

Ax)= Y..',. y{f~,(x)},

(lo)

where w, = weight factor of the k th partial goal function, fp~(x) = kth partial goal function. Weight factors express the relative importance of a particular objective, while another function, F, operating on a particular goal function, provides freedom to adjust the goals. Suppose the minimum of the real functionf(x) is to be found, where x is an n-dimensional vector within the prescribed region E s R" defined by equation (! !). g,(x) 1>0. i = l , p ,

h,(x)=O, j = l , r .

(11)

The problem of finding the optimum of the function f(x) within region E can be solved by determining the optimum value of the new function F(x) =f(x) + P(x),

(12)

in the unconstrained region R'. The penalty function P(x) can be formulated as follows: P

P(x) = r(p). ~ [min{0.g,(x)}[" + ~ Imin{0, hi(x)} + max{0, hi(x}[p,

( ! 3a)

1--1

tml

where rt p are real parameters, p is a real variable and r(p) a continuous real function such that:

r(p k) > r(p *÷') > O, lira r(p k) = 0.

(13b)

k~+ao

With the usual choice of r(p), a suitable total objective function F(x), materializing the penalty effect of equation (13) becomes:

(14) and can be considered defined over the entire R" space. In order to avoid an optimum outside the constrained region due to too small a penalty [equation (12)], the following procedure has been used: if in two subsequent base points of the search, xl and xl+ F(x~+, ) < F(x,),

Kinematic synthesis of m e c h ~

559

and

P(x,.,) > P(x,),

(15)

the search generates an inadmissable optimum, which implies that the penalty is not "strong enough". The parameter k (or ~,,/~) needs to be increased, until equation (15) is no longer satisfied when the last step in the search from xt to xt. t is repeated. 5. CREATION OF THE OBJECTIVE AND CONSTRAINT FUNCTIONS Total objective function

The goal function is a compound weighted goal function, assembled from partial goals representing geometric constraints. In a further stage of the design, kinematic goals (velocities, accelerations, jerks), as well as dynamic goals (forces, inertial effects) would be included in the total objective function as well. The most significant partial objective function expresses the percentage of the windshield covered by the wiper sweep. The windshield is assumed planar as in Fig. 9, in which A, = total area of windshield, A, = area of one segment. The function A,=A.-~A,

is to be minimized, where 5

,4.=~ ~. (x,-x,.,).(y,+ y,,,).

(16)

I--|

The second partial objective function measures the distances from the wiper blade tips to the nearest windshield edges. The sum of these distances (possibly weighted for special requests) is to be minimized. The respective distances between point P2(x2,y2) and the edges B C and C D are: dl =

x2h

+ y 2 ( b - l ) - bh Jh

2+

(b

,

d: = h - y2.

(17)

-- t ) 2

The next partial objective is related to the compactness of the entire mechanism. This is measured both by the maximum linear dimension of the housing of the mechanism and the area occupied by the windshield-wiper mechanism in the plane of motion of the blade. Constraints

The design constraints for the windshield-wiper mechanism basically express the fact that the wiper blade should be entirely contained within the windshield area at any instant, i.e. the blade must not intersect the windshield edges.

to D

c (l,h)

P

tOA

--x

B (b.ol

Fig. 9. Windshield [geometry (right half).

560

DAMIR VUCINA and F E g D ~ D

FI~UDENS'TEIN

For edge AB: y, i>0

(constr. No. 1),

(18)

y:~>0

(constr. No. 2).

(19)

For edge CD: (h - y : ) > / 0

(constr. No. 3)

(20)

For edge BC:

yp-y:>~O or

(h

• b - h • x:

b--t

)

y, >t0

(constr. No. 4).

(21)

/

In addition to these general functional constraints there are additional design constraints involving link size and other design parameters--for example see equation (27). 6. DIMENSIONAL SYNTHESIS AND O P T I M I Z A T I O N The kinematic structures which have been generated need to be analyzed further with regard to their ability to optimize the just listed functional requirements. Since the objective functions concern overall linkage performance, complete continuous simulation of linkage and wiper motion with the corresponding goal function is needed, in the case of structure 4 (Fig. 5) the following analysis applies.

Design variabh's d= OA,

f=ac, e=OB, I= El), P= EC, ~(initial). Synthesis requirements Variation of angle [3. ff'-180 ~, counterclockwise (CCW) rotation positive. Variation of angles ~ and ~. • (IJ = o °) = ~o, ~(# = o °) = ~o,

a(• = 90 °) = 270 °,

¢ ( # -- 90 °) = 90:.

The above specifications are based on: (i) The symmetry of wiper motion about the y-axis of the windshield (Fig. 9). (ii) The configuration in which the outer tip of the wiper blade is closest to the center of the sun gear occurs at the vertical position of the blade (/~ = 90°). (iii) In the initial position, the blade is horizontal (angle # = 0). Given the above specifications, angle ¢0 is determined as

• 0= ~(# =0°. ~' --~o) (see Fig. 5) with tan # = Ya -y.c = (d + e)sin ~ - f s i n X 8 m XC (d + e)cos q~ - f c o s ~,"

(22)

f sin ~o sin ~° = d + e

(23)

whence

Kinematic synthesis of mechanisms

561

The gear ratio, n. is determined as follows: (~t - %) = (! + n ) . ( ~ -- ~b0), n

=

T3 1 8 0 - ~ + ~ o = 7",. 90° - ¢o

--

(24) (25)

.

where 7"., and T3 denoting the number of teeth of the corresponding gears. Finally. x o = Xc + ( P + Y o = .Vc + (p +

1) cos/~,'[ I ) sin/~. ~

(26)

To prevent angle fl from becoming negative with increasing ~b, the condition f . (sin • - sin xo) - d- (sin 0o - sin ~b) ~
(27)

needs to be satisfied in the initial position of the mechanism, recognizing that for any current values of the variables angle % must satisfy [equation (28)]. Differentiating [equation (27)] with respect to a yields:

f

d +---e "d(sin ~c)l=-,0 ~
= (l + n). d,/, and

J' cos ~,. ( I + n) _< (a----T-e-3: c-o; I

(28)

A similar ~ t of equations is dcrived for this linkage for the ca.~ of internal gearing. The methodology outlined has l~cn applied to all structurcs, yiclding the necessary equations for the creation of the goal function and constraint subroutines in the dimensional optimization code. As a final result, the corresponding optimal dimensions as well as instantaneous and global values of the goal functions for each candidate mechanism were obtained. The motion simulation is shown only for thc optimally proportioned structures 4 and 5 (Fig. I0). With 89.9% of windshield area covered structure 5 is optimal from the point of view of efficiency. Note that the program code allows for any combination of relative weights of partial objectives. (a)

MECH 4 76.5 %

,,t\ MECH 5

\" Pt

89.9 %

Fig. 10. (a) and (b) Motion simulation of the wiper blades of the optimized mechanisms from Fig. 5 (structure 4) and Fig. 6 (structure 5) shown in the windshield frame.

Fig. i I. Guidance of a point on the wiper blade link via a planetary 6ear pair.

562

DAMIRVUCINAand FERDINANDFR,EUDENSTEIN 7. D I M E N S I O N A L S Y N T H E S I S OF W I N D S H I E L D - W I P E R VIA THE P R E C I S I O N P O I N T A P P R O A C H

MECHANISM

This approach was applied alternatively to the direct method just shown. Cycloidal Burmester theory was used for proportioning the geared-five-bar linkages, with the dyad modelling technique for generating a planetary gear pair. Let the n positions (Ps, as) of the body to be guided by defined (Fig. II) by

e,;

j = l. , .

al; j = l , n . where the vectors a/, Pj define a point and a line fixed in the body. One of the points of the floating link can be generated as the point of concurrency generated by the wiper blade assembly, while the other point is guided by a planetary gear pair on a cycloidal curve. The problem considered here is the synthesis of the planetary gear pair for this motion-generation problem. The specified n positions of the point P of the moving body are defined relative to a fixed coordinate system by the vectors R j . j = i, n. In the same coordinate system, the center of the fixed sun gear and the center of the planetary gear in the n positions (which are to be determined), are located by vectors m and k;, j = I, n respectively. Let the vector ws locate the center of the planetary gear in thejth position relative to the center of the fixed sun gear M. The vector g, determines the location of the pivot-point, C;, relative to the center, K. of the planetary gear. Finally, the vector z~. embedded in the moving body connects the given point, P . to the pivot-point, C/, on the planet. The problem can be considered solved if the vectors k~, g~ and z, are determined. Now m=R t-z,-gt-w;

and kl = m + w t .

(29)

The loop-closure condition for positions ! and j (Fig. II) is: %+g,+z~-6,-z~-ga-w~

=0.

(30)

w = wt. w, = w • exp(i//j). "~ g = gt, gj = g" exp(iT/).

(31)

Let

z=xj.

z1=z'exp(i~,).

Using the displacement equation of external planetary gearing with a fixed sun gear:

.13,.

(l +

(32)

Equation (30) becomes: w • (exp(ipj) - I) + g • (exp(inp) - l) + z • (exp(i:,s) - 17 = 6/.

(337

By appropriate choice of parameters, equation (337 can be specialized to simpler cases. For example, g = 0 yields the classical Burmester theory (four-bar linkage synthesis). In order for the solution (w. g. z) to exist, the "compatibility conditions" [equation (34) and others] need to be satisfied by angle Ps and the available parameters. In the case of the present five position synthesis four complex equations with three complex variables (w. g. z) exist, so that the augmented matrix of the system now has to be of rank 3. i.e. detlexp(i/1,)-!

exp(in,,)-I

exp(icx,)-I

k - - 2 ..... 5

3,1 --0.

(347

8. C O N C L U S I O N

This case study has once again established the power of graph theory for kinematic (type) synthesis. Nonlinear programming methods--in particular multivariable numerical optimization in

Kinematic synthesis of mechanisms

563

constrained regions by means of penalty functions--have been found to be extremely valuable for the task of dimensional synthesis. However, the precision-point approach to dimensional synthesis did not prove advantageous in this design; since the design criteria relate to overall system performance and there are a variety of constraints, so that several restarts with distinct input data (positions specifications and free choices) are needed. Thus the former method seems preferable. The design, methodology and approach which have been outlined in this investigation, has yielded several new and potentially useful automotive windshield-wiper configurations. Acknowledgement--The project, portions of which are described here, has been part of a study program at Columbia University in the City of New York sponsored by the Institute for International Education. New York, N.Y., U.S.A., by providing a Fulbright Grant for the first author. REFERENCES I. F. Freudenstcin and E. R. Maki. J. Environ. Plann. 6, 375-391 (1979). 2. Anonymous. Automot. Engng 93(7). 67-68 (1985). 3. G. N. Sandor and A. G. Erdman, Advanced Mechanisms Design. Analysis and Synthesis. Vol. II. Prentice-Hall, Englewood Cliffs, N.J. 0984). 4. H. N. V, Temperley, Graph Theory and Applications. Wiley, New York (1981). 5. R. l. Alizade, A. V. Mohan Rao and G. N. Sandor, Trans. ASME JI Engng Ind. 97, 629-634 (1975). 6. F. Freudenstein, Trans. ASME JI Engng Ind. IlIB, 15-22. (1959). 7. F. Freudenstein and E. R. Maki. Trans. ASME JI Mech. Des. 105. 259-266 (1983). 8. M. Mayourian and F. Freudenstein, Trans. ASME JI Mech. Tra~Tm. Automn Des. 106, 458-461 (1984). 9. D. G. OIson, A. G. Erdman and D. R. Riley, Mech. Mach. Theory 20, 285-295 (1985). 10. E. J. F. Primrose, F. Freudenstein and G. N. Sandor, Tran.~. ASMEJI Appl. Mech. Dec. 683 693 (1964). I i. D. Rosen, D. Riley and A. Erdman, Des. Engng Tech. Canf Columbus, Ohio. Oct. (1986). 12. J. Petric and S. Zlobec, Nelinearno Programiranje. Naucna Knjiga, Beograd (1983). 13. D. Vucina, Master's Thesis, Columbia University in the City of New York, Oct. (1987).

ANWENDUNG VON GRAPHTHEORIE UND NICHTLINEAREM PROGRAMMIEREN BEI DER KINEMATISCHEN SYNTHESE VON MECHANISMEN Zmmmm,mfmsmng~Es wird eind systcmatische "'Anfang aus Nichts" Mcthodc zum Projcktieren yon Mcchanismcn bcschriebcn und angcwendt bcim Entwerfen yon cinem Automobil.vchcibcnwischer. Folgcnd der Spccifikation yon Designkriterien und Einschriinkungen, die m6glichcrweise nutzbarcn kincmatirchen $trukturen werden gcnericrt mi! Hilfe yon Graphthcorie. Einc funcktionale Analyse yon dicsen Strukturcn licfert einigcn brauchbarc Mcchanismen. Nach der Typsynthcse folgt als tin Nichtlineares-Programmieren Problem aufgcstellte Dimensionalsynthcse. Die Ziele und Bcschr~nkungcn bci diesem Design werden in Ziel- und Pcnaltyfunktionen transformiert und danach optimiert. Das Endcrgebnis ist cin dimensionierter optimaler Mcchanismus. Die alternative Methode dcr Dimensionalsynthcs¢ durch Precision-Point Synthcse wird auch diskutiert.