Kinematic optimization of mechanisms: A finite element approach

Kinematic optimization-of mechanisms,

a .finite'element approach .

Kinematic optimization of mechanisms,

i^ a finite element approach

Kinematic optimization of mechanisms,

a finite element approach

Proefschrift

Ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, Prof.dr. J.M. Dirken, in het

openbaar te verdedigen ten overstaan van een commissie door het College van Dekanen daartoe

aangewezen, op 13 oktober 1987 te 14.00 uur door

Antonius Johannes Klein Breteler, geboren te 's-Gravenhage, werktuigkundig ingenieur.

Prof. Dr.-lng. H. Rankers

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Klein Breteler, Antonius Johannes

Kinematic optimization of mechanisms, a finite element approach/Antonius Johannes Klein Breteler. - Delft: Technical University

Proefschrift Delft. - Met lit. opg. - Met samenvatting in het Nederlands.

ISBN 90-370-0012-6

SISO 650 UDC 531.1:621.01 (043.3) Trefw.: stangenmechanisme; kinematica.

Aan Ria,

In mechanical engineering the design of link mechanisms is recognized as a difficult problem with many theoretical and design aspects. Kinematic dimension synthesis is one of these aspects: the calculation of the kinematic parameters for a given motion problem. Basically a kinematic synthesis problem can be formulated as a set of (nonlinear) equations for these parameters for which various solution methods, including numerical methods and optimization, may be used.

The report describes the theory to define and to solve such problems in a general way, employing a certain approach of the Finite Element Method (FEM) and appropriate optimization techniques. The feasibility of the kinematic parameter space is treated extensively. It is demonstrated that singularity of the continuity matrix in the FEM yields an exact expression for the boundary. For optimization purposes all derivatives, required to calculate

the gradient of both the error function (least squares type) and the boundary function, are expressed exactly in terms of the FEM definition of transfer functions.

The theory has been programmed to handle almost any kinematic synthesis problem reliably, efficiently and accurately; the definition of both the mechanism and the motion problem is done "at input level" by means of tables

(no formulas required). Several synthesis problems are reported which have been solved successfully with the program.

ACKNOWLEDGEMENT

This investigation was carried out in the Laboratory for Automation of Production and for Mechanisms of the Delft University of Technology, where the author is a member of the scientific staff.

The author wishes to thank all students and colleagues who cooperated directly or indirectly to make this result possible.

Sincere thanks are due to the members of the workgroup CADOM (Computer Aided Design Of Mechanisms), in which the research activities on mechanism design are concentrated, and to its inspiring founder Prof.Dr.-Ing. H.Rankers. The author owes much to Prof.dr.ir. J.F.Besseling, Prof.dr.ir. J.J.Kalker and Prof.dr.ir. K. van der Werff for their critical remarks and help with specific details, and to J.Booij M.Sc. for his help with the english language.

-0.1-CONTENTS

List of symbols and abbreviations 0.3

Samenvatting (summary in Dutch) 0.5

1. INTRODUCTION 1.1 1.1 Kinematic optimization in the design process of mechanisms 1.1

1.2 Characteristic optimization problems in kinematics 1.2 1.3 Generality of optimization as a synthesis method 1.6

1.4 Purpose of this research work 1.8

2. KINEMATIC MOTION ANALYSIS BY THE FINITE ELEMENT METHOD 2.1 2.1 The FEM in the design process of mechanisms 2.1 2.2 Description of elements: the continuity equations 2.2 2.3 Mechanisms and their generalized first order

transfer functions 2.5 2.4 Iterative calculation of the zero order transfer function 2.9

2.5 Calculation of the second order and other transfer functions 2.11 2.6 Summary and conclusions concerning the FEM-approach

for kinematic analysis 2.13

3. KINEMATIC MOTION QUANTITIES DEFINED BY TRANSFER FUNCTIONS 3.1 3.1 Single transfer function values and motion quantities

in one mechanism position 3 • 1 3.2 Motion quantities concerning more than one

mechanism position 3-2 3.3 Kinematic motion: time dependent or time independent 3-2

3.4 Motion quantities concerning multi degree of freedom

mechanisms 3-5 3.5 Prescription of over-determined motion for the degrees

of freedom 3.7 3.6 Summary and conclusions concerning motion quantities defined

by transfer functions 3-10

4. BOUNDS OF THE KINEMATIC PARAMETER SPACE 4.1 4.1 Positive length of the binary element 4.1 4.2 The determinant function det D = 0 as boundary function 4.2

4.3 The gradient of the determinant for det Dc * 0 4.6

4.4 The gradient of the determinant for det D = 0 : definition

with maps 4.7 4.5 Calculation of the normal direction of the boundary function

det Dc = 0 4.10

4.6 Behaviour of det D near the boundaries 4.13 4.7 Special cases of det D boundary behaviour 4.16

4.7.1 Problems with a zero value for the second order

transfer function 4.16 4.7-2 Problems concerning undefined transfer functions 4.17

4.8 Non-concavity of the determinant function det D 4.18 4.9 Summary and conclusions concerning the bounds of the

5. THE (UNCONSTRAINED) ERROR FUNCTION 5.1 5.1 Definition of the kinematic optimization problem 5-1

5.2 Gradient, stepsize and Hesse-matrix 5-3 5.3 Considerations concerning the optimization strategies 5-6

5.4 Under-determined optimization problems 5-7 5.5 Summary and conclusions concerning the error function 5-9

6. THE CONSTRAINTS 6.1 6.1 Bounds of the feasible kinematic parameter space 6.1

6.2 The range of individual kinematic parameters 6.2 6.3 Conditions between kinematic parameters 6.3 6.4 Tolerance and range of motion quantities 6.5 6.5 Summary and conclusions concerning the constraints 6.7

7. OPTIMIZATION STRATEGIES 7-1 7.1 General considerations 7-1 7.2 Penalty function strategy 7-3

7.2.1 General explanation 7-3 7.2.2 The constraint det D * 0 as interior penalty function 7-4

7.2.3 The length of a binary element as an interior

penalty function 7-6 7.2.4 Other constraints: exterior penalty functions 7.6

7.3 Gradient projection method 7-7 7.3-1 General explanation 7 • 7

Q

7.3.2 Gradient projection on the boundary function det D = 0 7-9 7.3.4 Other application of the gradient projection method 7-10

7.4 SQP/ variable metric method 7.11 7.4.1 General explanation 7-H

7.4.2 Application 7.12 7.5 Conclusions and final choice for the best optimization

strategy 7-12 8. PROGRAMMING CONSIDERATIONS 8.1

8.1 Ordering of variables and residuals 8.1

8.2 Program structure 8.3 8.3 The user program LSQFUN 8.5 8.4 Computer requirements 8.7 .8.5 Summary and conclusions about the programming concept 8.9

9. EXAMPLES 9.1 9.1 Four-bar linkage with prescribed stroke of the rocker angle 9-1

9.2 Boundary behaviour of an inverted slider-crank mechanism 9-4 9.3 A six-bar linkage (Watt-2) generating a coupler curve with

an approximated straight line and constant speed 9-6 9.4 A six-bar linkage (Stephenson-2) generating a rotational

sine function 9-H 9.5 Adjustable stroke in a combustion engine 9-14

9.6 Summary and conclusions concerning the examples 9-21

10. SUMMARY AND CONCLUSIONS 10.1

11. REFERENCES 11.1

APPENDIX A: DESCRIPTION OF THE KINEMATIC ELEMENTS A.l APPENDIX B: LIST OF KINEMATIC MOTION QUANTITIES B.l

-0.3-LIST OF SYMBOLS AND ABBREVIATIONS

A,B,C,D,E,F: Point of a mechanism

a,b,c,d,e,f: Distance between two mechanism points

i,j,k,l,m,n: Subscript for coefficient numbering in vectors, matrices etc. B Barrier term in an exterior penalty function

c Superscript for dependent coordinate x. D 1. Matrix of first order continuity equations

2. Operator to denote a derivative map D Kinematic map X--» E

det Determinant of a matrix

d 1. Operator to denote differentiation of a single-valued function 2. Number of degrees of freedom of a mechanism

E Vector space of form parameters e

F 1. Objective function (error function) to be minimized 2. Vector space of applied forces in the FEM

FEM Finite element method

f 1. Vector of generalized applied forces in the FEM

2. Residual function in the sum of squares F to be minimized G Desired motion value (goal value)

g 1. Inequality constraint function

2. Form parameter in a pair of gears or a gear-and-rack H Hesse matrix of second partial derivatives 32F/3v3v

h Equality constraint function

i Mechanism parameter: transm. ratio of a pair of gears or a gear+rack I Identity matrix

k 1. Superscript, element number 2. Order of a motion problem ker Kernel of a map (matrix)

I Length of a binary link

L Lagrangean function

M 1. Actual motion quantity, generated by a mechanism 2. Centre of curvature of planar motion

m Superscript denoting prescribed motion, as used for nonzero deformations Ac and displacements Ax

(k)

m: Number of discrete motion conditions of order k in mechanism position i N Number of residual functions f (motion conditions)

n 1. Number of parameters, functions etc. 2. Order of a gradient method

c c p c n Dimension of vector x , vector e and matrix D

n. Dimension of extended Lagrangean space (k)

n: Number of input motion quantities of order k, including time derivatives, in mechanism position i

o Superscript for prescribed zero deformations or displacements

p 1. Superscript for prescribed form parameters E , either zero or nonzero deformations Ac

2. Subscript, number of positions of a mechanism P Inverse kinematic map E ■» X

P,Q End-points of a binary element

r 1. Controlling parameter to weight penalty functions 2. Superscript for dependent form parameter E

R Instantaneous centre of rotation in planar motion SQP Sequential quadratic programming

s 1. Position of a slider

2. Curve vector in planar motion 3. Seach direction in optimization

t 1. Time

2. Vector with directional derivatives of a hyperplane u 1. Lagrange multiplier for inequality constraint

2. Form parameter used in a ternary element v 1. Vector with kinematic parameters

2. Form parameter used in a ternary element

V Vector space of all kinematic parameters v to be optimized w 1. Weight factor for a discrete motion condition

2. Lagrange multiplier for an equality constraint X Vector space of coordinates x

x 1. Coordinate vector 2. Single coordinate in xOy y 1. Single coordinate in xOy

2. Dependent part of vector v, as used in the gradient proj. method z Independent part of vector v, as used in the gradient proj. method a 1. Crank angle

2. Any prescribed form parameter or coordinate {} 1. Rocker angle

2. Angle of element (either coordinate or form parameter) T Coupler angle

6 Kronecker delta

3 Operator for partial differentiation

A Operator for finite or infinitesimal variation e 1. Vector of form parameters

2. Single form parameter

A 1. Mechanism parameter: crank length divided by frame length 2. Vector with generalized mechanism parameters

K Mechanism parameter: coupler length divided by frame length <t> Direction of planar motion of a point

* Rigid angle between two kinematic elements p Radius of curvature

0 1. Vector with generalized stresses (internal forces) in the FEM 2. Mechanism parameter: rocker length divided by frame length 1 Summation operator

9 Rigid angle with a crank (assembly parameter)

V Operator for differentiation of a multivariate function

' 1. Operator for differentiation of a single or generalized transfer function with respect to the degrees of freedom E of a mechanism 2. Idem with respect to all prescribed form parameters e

" See at ', but now second derivative

Operator for differentiation with respect to time Operator for second derivative with respect to time

, In index notation: operator for differentiation + subscript separation * 1. Superscript, involved with a (nearly) zero pivot of matrix D

2. Superscript, involved with a dual or alternative mechanism 9 Direct 3um of vector spaces

U Union of vector spaces 6 Member of a vector space

<,> Inner product of two vectors from spaces with different-physical dimensions

Index notation: repeated subscripts.in a multiplication indicate summation: a.b. = I a.b.

-0.5-SAMENVATTING : "kinematisch optimaliseren van stangenmechanismen, een eindige elementen aanpak."

Vraagstukken op het gebied van kinematische synthese van stangenmechanis­ men kunnen in principe worden opgelost met behulp van numerieke methoden, zoals optimaliseren. Gebruikelijk hierbij is om het bewegingsvraagstuk uit te drukken in discrete bewegingsvoorwaarden, welke gewoonlijk een niet-lineair stelsel vergelijkingen in de kinematische parameters vormen. De af­ wijkingen ten opzichte van de voorgeschreven bewegingsgrootheden, de resi­ duen, worden gewogen, gekwadrateerd en opgeteld. Tesamen vormen zij de fout-functie (objectfout-functie), welke geminimaliseerd moet worden. Het stelsel ver­ gelijkingen verschilt echter per bewegingsopgave en per type mechanisme. Indien een gradiëntmethode voor het optimaliseren gebruikt wordt, zijn de partiële afgeleiden van de foutfunctie naar de kinematische parameters nodig. Omdat het verkrijgen van de exacte uitdrukkingen, voor elk type mechanisme en elk bewegingsvraagstuk opnieuw, moeilijk en tijdrovend is wordt optimaliseren in de praktijk weinig gebruikt door ontwerpers van stangenmechanismen.

Door de komst van de digitale computer ontstonden algemene methoden om mechanisch gedrag te beschrijven. Zo werd de eindige element methode (FEM)

toegepast om de kinematische en dynamische analyse van een gegeven mechanis­ me te formuleren [6]. Dit onderzoek ontwikkelt deze aanpak verder teneinde kinematische synthese en optimalisering mogelijk te maken, waarbij alle ver­ eiste vergelijkingen en afgeleiden automatisch worden berekend. Het bewe­ gingsvraagstuk kan aldus "op invoerniveau", d.w.z. in tabelvorm, aan een rekenprogramma worden aangeboden.

Dit verslag beschrijft eerst kort de FEM-aanpak van kinematica. De sleu­ telrol in de theorie wordt gespeeld door de eerste orde continuïteitsverge­ lijkingen, welke de gelineariseerde betrekkingen tussen de bewegende coördi­ naten en voorgeschreven vormparameters uitdrukken door middel van de matrix D . Het FEM-model van een mechanisme dient hierbij zodanig te worden opge­ steld, dat de voorgeschreven vormparameters de kinematische parameters voor­ stellen ("elke schakel is een element"). De beschrijvingen van een aantal elementen, welke veelvuldig worden gebruikt in vlakke stangenmechanismen, zijn in dit verslag verzameld en aangevuld.

De gebruikelijke bewegingsgrootheden waarin bewegingopgaven door ontwer­ pers worden geformuleerd, werden uitgedrukt in termen van de overdrachts­ functies (beweging van de coördinaten), welke het resultaat zijn van de FEM-berekeningen. Al deze uitdrukkingen werden gedifferentieerd naar de vormpa­ rameters om de gevraagde afgeleiden voor de gradiëntmethoden te verkrijgen.

De ruimte van voorgeschreven vormparameters, inclusief die welke de aan-drijfgrootheden voorstellen, is in deze aanpak de kinematische parameter-ruimte. Deze ruimte wordt verdeeld in gebieden waarin het mechanisme wel en niet kan bestaan, de grens tussen deze gebieden wordt gedefinieerd door het nul worden van de determinant van de matrix D . De ruimte waarin optimalise-ring moet plaatsvinden wordt gekarakteriseerd doordat det D hetzelfde teken moet hebben voor alle standen van het mechanisme waarin bewegingsvoorwaarden zijn voorgeschreven. De gradient van de grensfunctie det D = 0 heeft gewoonlijk oneindig grote coëfficiënten. De normaalrichting van de grens kan echter zeer eenvoudig verkregen worden uit het veegproces van de matrix. Aangetoond werd, dat de richtingsafgeleiden van het raakvlak aan de grens

cT

zijn bepaald als de kernvector van de getransponeerde matrix D . Desondanks is een optimaliseringsstrategie, welke precies de grens volgt, niet aantrek­ kelijk omdat de meeste bewegingsgrootheden op de grens niet gedefinieerd of oneindig zijn. Det D = 0 drukt tevens zogenaamde "nul-kwaliteit" van bewegingsoverdracht uit. Voor een strategie om de grens op enige afstand te

volgen werd de gradiënt van de determinant uitgedrukt in overdrachtsfunc­ ties, waarbij de eerste en de tweede orde overdrachtsfunctie nodig bleek.

Een tweede natuurlijke grens van de kinematische parameterruimte ontstaat door de definitie van het binaire kinematische element, welke geen lengte

B. £ 0 toestaat. Theoretisch is deze grens niet moeilijk te verwerken. Indien

voor het bewegingsvraagstuk een negatieve stanglengte van belang is, kan de ontwerper dit probleem vermijden door een ander element (ternair) te kiezen in plaats van het binaire element.

Optimaliseringsstrategieën gaan uit van de eerste en tweede orde partiële afgeleiden van de foutfunctie: de gradientvector respectievelijk de Hesse-matrix. Om deze te berekenen zijn overdrachtsfuncties van éèn respectieve­ lijk twee orden hoger dan de orde van het bewegingsvraagstuk nodig. Omdat bij de huidige stand van de FEM een hogere dan de tweede orde overdrachts­ functie niet beschikbaar is, moet een compromis worden gesloten tussen de kwaliteit van de optimaliseringsstrategie en de toe te laten orde van het bewegingsvraagstuk. Uit praktische overwegingen werd de aandacht gericht op gradiëntmethoden (Gauss/Newton, quasi-Newton), waardoor het nog mogelijk is om eerste orde bewegingsvraagstukken te verwerken.

Met betrekking tot nevenvoorwaarden werden een drietal bekende optimalise­ ringsstrategieën beschouwd om aan te knopen bij de FEM-berekeningen: de straffunctiemethode, de gradient-projektiemethode en de SQP (= sequential quadratic programming) methode. Met name de gradient-projektiemethode biedt een slimme mogelijkheid om de grens van de optimaliseringsruimte te vermij­ den, omdat het raakvlak aan (nabij) deze grens eenvoudig uit de FEM-aanpak volgt. Aan de straffunctiemethode werd echter de voorkeur gegeven omdat deze meer aansluit bij de ontwerppraktijk: door de ontwerper kunnen allerlei nadere beperkingen, welke vrijwel altijd als ongelijkheden worden geformu­ leerd, in rekening worden gebracht. Hierbij worden alle ongelijkheden be­ treffende de voorgeschreven beweging en de te optimaliseren parameters als extra residuele functies opgevat en toegevoegd aan de som van kwadraten. Het optimaliseringsprobleem wordt dus als een probleem zonder nevenvoorwaarden behandeld. De "natuurlijke" grenzen worden beschermd door een inwendige straffunctie om afstand ervan te kunnen houden. De door de ontwerper op te geven beperkingen worden bestraft na overschrijding ervan (uitwendige straffunctie), waardoor de oplossing enigszins buiten deze grenzen kan worden gevonden. Door de gewichtsfactoren van deze extra residuen kan de ontwerper het resultaat in de hand houden.

Een rekenprogramma werd geschreven dat, uitgaande van een willekeurig mechanisme en een willekeurig bewegingsvraagstuk, alle kinematische

bewegingsgrootheden en vereiste afgeleiden berekent, teneinde de foutfunctie en zijn gradient automatisch te bepalen. Dit rekenprogramma werd samen met een standaard optimaliseringsprogramma (Gauss-Newton methode) gebruikt om enkele karakteristieke kinematische syntheseproblemen op te lossen. Er kon worden vastgesteld, dat de automatische bescherming van de kinematische parameterruimte effectief en betrouwbaar functioneert. Door de numerieke aanpak vragen ingewikkelde mechanismen (meer dan 20 bewegende coördinaten) echter nogal wat rekentijd, terwijl ook enig significantieverlies in de berekeningen merkbaar is.

Voor de ontwerper van stangenmechanismen is het rekenprogramma een tot nu toe ongekend veelzijdig stuk gereedschap bij het oplossen van kinematische synthesevraagstukken. Omdat veel van het resultaat echter afhangt van het goed kunnen opstellen van een FEM-model van het mechanisme en het bewegings­ vraagstuk (inclusief de nevenvoorwaarden), is goed onderwijs en oefening een noodzakelijke voorwaarde voor een succesvolle toepassing.

Ondanks de veelheid van mogelijkheden, welke in dit verslag zijn genoemd, is het zinvol het onderzoek op het gebied van kinematisch optimaliseren voort te zetten. Nuttige uitbreidingen betreffen bijvoorbeeld:

- ruimtelijke kinematica: ruimtelijke elementen en bewegingsgrootheden, - de derde orde overdrachtsfunctie, waardoor ook tweede orde bewegings­

-1.1-1. INTRODUCTION

1.1 Kinematic optimization in the design process of mechanisms

Mechanisms are widely used in production machinery to perform motions or transmit forces in all kinds of production processes. Most of these mecha­ nisms are designed to displace tools in a machine, in order to handle or manufacture discrete products. Mass production requiring high speed machin­ ery, e.g. in packaging machines and textile machines, is a favourite area of application because of the good dynamic properties and reliable operation that can be achieved with link mechanisms. Other applications are found in positioning parts like a garage door, the adjustable seat of a wheel-chair and non-moving but adjustable tools in so called "flexible automation".

Machine parts to displace have three degrees of freedom in planar motion or six degrees of freedom in spatial motion. Application of a mechanism may reduce this number of degrees of freedom, thereby reducing the driving pro­ blem. Less costs, especially with respect to driving motors and control equipment, or easier (manual) control, can be a good reason to apply a mechanism as well.

The design of mechanisms is a complicated, highly iterative process with several design steps, see fig. 1.1. A global division can be made in "kine­ matics" and "dynamics", which is by definition a separation into the "rigid body motion" problem and the "force" problem. According to this design philosophy the rigid body motion problem has to be solved first, because kinematics implicitly play a role in dynamics.

Kinematic synthesis is widely recognized as the most difficult step in kinematics. It consists of firstly [27] the choice or determination of the mechanism type ("type synthesis") and secondly the calculation of the kinematic parameters of the chosen mechanism ("dimension synthesis").

Kinematic optimization deals with dimension synthesis: the kinematic parameters of a chosen type of mechanism are to be calculated in order to generate a certain desired motion in a machine "as nearly as possible". In problems where the number of kinematic parameters is greater than or equal to the number of motion conditions, so called under-determined or determined problems, the solution may be exact. In that case the calculation process can be considered also as a "(non-)linear programming" problem. The charac­ teristic property of optimization is however the ability to calculate the

parameters for over-determined motion problems such that minimal error will be generated with respect to the desired motion.

Concerning the types of mechanisms it can be remarked that both cam mechanisms and open-loop mechanisms ("industrial robots") must be excluded from the kinematic optimization idea, since the kinematic motion is expli­ citly programmed on an information carrier. Kinematic optimization is espe­ cially applicable to link mechanisms with single degree of freedom or few degrees of freedom and over-determined motion problems.

K I N E M A T I C S d e f i n i t i o n of m o t i o n p r o b l e m

£

Kinematic s y n t h e s i s : c h o i c e / d e t e r m i n a t i o n of mechanism t y p e and parameter values

Kinematic analysis: judgement of rigid motion

/ K i n , concept (model) ƒ F E M : - w i t h o u t forces, - p r e s c r i b e d (zero) d e f o r m . D Y N A M I C S i n t r o d u c t i o n of applied f o r c e s Dynamic s y n t h e s i s : c h o i c e / d e t e r m i n a t i o n of mass and s t i f f n e s s d i s t r i b u t i o n Dynamic analysis: judgement of i n t e r n a l forces and d e f o r m a t i o n s

31

/ f i n a l d e s i g n F E M :

with forces & deformations

Fig. 1.1 Finite Element Method (FEM) and the design process of mechanisms.

1.2 Characteristic optimization problems in kinematics

A classical group of synthesis problems is depicted in fig. 1.2. The desired motion is assumed to consist of discrete conditions, either

- angular coordination between two links that can rotate in the same plane (perform planar rotation), or

- point position coordination of a plane with respect to rotary input mo­ tion, or

- coupler motion of a plane.

As is generally known these motion problems can be solved exactly for at most 5 discrete conditions of the same type, choosing the four-bar linkage. The solution method is either analytical [1] or graphical

-1.3-as an optimization problem. The-aim-of the kinematic synthesis is to calcu­ late all kinematic parameters A. such that the coupler curve of coupler point C passes through N specified points x„., y„., with i = 1 N and G

01 Ui as the index for desired motion (goal value).

Assuming all crank angles a. as given input motion quantities, at most 10 independent kinematic parameters can be defined. Referring also to figs. 1.3 and 1.4 for a detailed description, they can be chosen as:

Aj = { xA o ' yAo' XBo' yB o ' a' b' C' e' f' 9 )

Denoting index M for any actual mechanism function, xM and y„ are functions of a with kinematic parameters A.. Now the non-linear system of equations to be solved can be written as:

XGi = xM( Aj 'ai)

yG i " yM (*j'a±)

i =1. •

For N Ï 5 the problem can be considered as a least-squares optimization problem to minimize the objective function F:

N

F(A.) = . n < x

M

- x

G

)

2 +

<y

M

-y

G

)

2

).

_(1.1)

V

t

A l

D

IT M : ©■"""^

Ao

_ B

-^C

X a) angular coordination c) 2

3 - , -1-4

*|—!-^h—I

a\~a

2

al a

A

\

n

yr

point position coordination

- * i

x3 H x5 — - X

coupler motion

■a

_yr

Fig. 1.2 Kinematic motion definition. Three classical types of discrete

conditions.

I t makes quite a difference if the crank angles a. are not specified, because in that case they also become subjects of optimization. The objec­ tive function is then to be modified to:

F(A.,a.) = . n < x

M

- x

G

)

2 +

( y

M

- y

G

)

2

} .

(1.2)

in which of course the constant angle 6 in the crank plane ("assembly parameter") has to be dropped. The new (vector of) kinematic parameters con­ sists of:

A. = {x. , y. , x0 , y_ , a, b, c, e, f} .

j l Ao JA o Bo JBo

It can be recognized that the vector space of kinematic parameters is in general bounded: e.g. the link length b in the example cannot be increased unlimitedly without disconnecting the linkage and making the objective func­ tion F undefined.

Very often motion problems can adequately be described using derivatives of motion functions. As an example the angular stroke of a rocker angle B

(see fig. 1.3) can be considered. Prescribing the stroke B - B . as a "max 'min discrete motion condition the three equations

stroke transfer function p t X ^ a )

kinematic parameters \ j = { A , o , x , e , < | > } X = a / d o = b/d x = c/d prescribed s t r o k e , conditions: 1' Pmax-Pmin 2) P ' ( X j / ! „ , „ ) = 0 3) 3 ' ( Xj / a m i n) = u

2Tt a

-1.5-(P- min G HM j max' ïM j min M(A ., a . ) dg.

da 3' (A ., a ) M J max dg.

da W,(A., a . ) MM j m m

must be fulfilled for calculable values of A ., a and a . . Supposed that j max min

other angular coordination values a-,.,^-,., i=l,...,N are prescribed, and ui ui

that the kinematic parameters A. consist of A. = { A , O ,

*}

the least-squares objective function can be expressed as

F(A., a , a . ) = I OM

j max m m ._ VKM Ï )

V i

2 ■ {(p -max m m M f> . )„ -%}max + ^ m i n

pmin'GJ

In engineering practice the so called "pick-up motion" of a coupler point is recognized as a difficult synthesis problem. This (planar) motion is depict­ ed in fig. 1.4. Coupler point C must not only have a cusp in its coupler curve, but the cusp tangent is also a prescribed motion quantity, usually

kinematic parameters collineation

\_-axis

V

{:

CAo ' yAo ' XB o ' yBo a

, b , c , e , f }

Ao Jk coupler . \ curve

inst. centre of rotation P s C

desired curve property: cusp

9

c

=9<r

object

W/W//MSWM

prescribed pick-up behaviour, conditions: 1) Xc(Xj,(Xc) 2) yc(Xj /ac )

3) xJ.(Xj,a

c

)=0

4) y'cUj,a

c

) = 0

5) X c l X j . O c ) = 0

perpendicular to the pick-up plane. As is generally known in kinematics a cusp is generated by a coupler point when it coincides with the momentaneous centre of rotation of the coupler. Normally the tangent direction * to a curve can be expressed by tan $ = r~-, or alternatively, using the first

or-dv dx Y '

der transfer functions y' = -r~ and x' = -r—, by tan 4> = •'-r. In the pick-up

J da da J x'

problem however both x' and y' are zero in the considered position a , and consequently the division has an undefined result. Now the tangent direction can be defined using the second order components of the transfer function: tan * = ^-. The pick-up problem can then be described by the following five equations: x

Gc

= x

M c

(

V

a

c

) y

Gc

= y

M c

(

V

a

c

) x

Gc =

x

Mc<V

a

c> = °

y

Gc =

y

M c

(

V

a

c> = °

y„ (A., a ) * .. Mc j c *_ = arctan „ ,,■' . Gc x" (A ., a ) Mcv j c

These examples show clearly that kinematic synthesis problems may contain various types of motion conditions, which can be prescribed in various combinations.

Optimization has the capability to be a general synthesis method because the calculation process can handle all combinations of motion conditions ("mixed conditions"). This is perhaps the most attractive aspect of optimi­ zation , and its characteristic advantage over other synthesis methods, which explore motion conditions of the same type ("uniform conditions"). This generality aspect will be discussed further in the next chapter.

1.3 Generality of optimization as a synthesis method

Generality with respect to kinematic synthesis of mechanisms must be understood as generality with respect to the following two aspects:

- the type of mechanism, and - the type of the motion problem.

In kinematic optimization these two aspects can be recognized easily. For instance in the least squares sum of residual functions f., concerning all motion conditions:

N N

F = I (f.r = I (M - G>2 (1.3)

-1.7-each residual depends on both the type of mechanism and the type of motion quantity M (M is to be compared with its desired value G ) . Examples of such motion problems were given in the previous chapter.

Comparable problems have been defined and solved by many researchers in the last decades, utilizing the facilities of modern digital computer equip­ ment and general purpose optimization software programs [17. 18, 20, 21, 23, 33. 34. 35. 36, 43 through 62]. Optimization is however not a single tech­ nique. A confusing variety of optimization methods exist, which can globally be distinguished in "direct search" methods and "gradient" methods. The gra­ dient methods explore the gradient vector, which is by definition the vector of partial derivatives of the objective function F with respect to all para­ meters to be optimized. These methods are widely accepted as more effective, reliable and accurate for medium size problems (10 - 50) parameters, which is also the scope of kinematic optimization [4].

It is common practice in kinematic optimization to define the objective function using analytical expressions for the residual functions. This means however that each motion problem, and each candidate type of mechanism, con­ stitutes a separate optimization problem for which all formulas and deriva­ tives must be supplied. Especially in mechanism synthesis these formulas tend to be complicated, even for very simple mechanisms. In such practice however researchers too often are not inclined to provide all formulas actually needed. The generality that optimization could offer was sacrified by choosing "uniform conditions" or focussing on only one type of mechanism. Sometimes this generality was partially saved by choosing a direct search optimization method, a less efficient and less accurate method, which avoids the formula derivations [33, 34, 35, 53, 65].

In kinematic synthesis it would be very valuable if the designer could have the opportunity to play a little bit with the definition of the motion problem and his ideas of candidate mechanism types [5]. This would not only challenge all those intuitive aspects of the design philosophy, but also make the iterative character of the design process more practicable. Obviously generality of optimization as a synthesis method can only be achieved if the designer can be freed from the need for formula derivations. In the present state of technique there is only one way this can be done adequately: the residual functions, and hence the transfer functions of the mechanism to consider, must be calculated numerically.

In [6,7] a certain approach of the so called finite element method (FEM) has been described as a numerical method to calculate kinematic (and

dynamic) motion quantities. In [8] the FEM applied to kinematics has been investigated with respect to kinematic optimization. Calculation of the

gradient VF of the objective function, defined by its derivatives to any variable v. to optimize:

,„ N 3M.

V F = f = 2 1 ( M - G ) . ^ (1.4) j i=l J

requires the numerical calculation not only of the motion quantity M, but also of its partial derivatives with respect to v.. It was concluded from [8] that the derivatives, as necessary in the iterative calculation of mo­ tion quantities (kinematic analysis) can immediately be exploited to calcu­ late the gradient of the objective function F. So a fruitful combination of FEM and kinematic optimization is the underlying idea of this reseach work.

1.4 Purpose of this research work

The purpose of this work is to describe the kinematic optimization theory such that generality is achieved with respect to both the type of mechanism and the type of motion problem. As the FEM applied to kinematics is expected to contain this generality, the integration of FEM, optimization and kinematic synthesis has been chosen as the research field in this work.

This research field has several aspect that will be dealt with. Of course the FEM-description of kinematics, originally described in [6], will be recapitulated and, where necessary, extended for optimization purpose. As no general description of boundaries in the kinematic parameter space could be found in literature, much attention has been given to this subject.

From the designers point of view it is interesting to have a survey of all possible motion conditions. Often motion problems do not only contain "equality conditions of motion" expressed by M - G = 0 (like the character­ istic examples in chapter 1.2), but also "inequality conditions of motion" of the type M - G è O o r M - G S O . This reflects however immediately upon the possibilities of the various optimization techniques.

Both the FEM method and the optimization methods require the use of the digital computer. The theory to be described in this work has been imple­ mented in a computer program being developed at the Laboratory for Automation of Production and for Mechanisms of the Delft University of Technology. The detailed user information of this computer program is not reported here, but the global structure, present capabilities and some characteristic results are presented to complete this research work.

-2.1-2. ANALYSIS OF KINEMATIC MOTION BY THE FINITE ELEMENT METHOD

This chapter summarizes the finite element approach as applied in [6, 7. 9] for the kinematic analysis of mechanisms. The use of transfer functions is explained to define those motion quantities, which play frequently a role in analysis problems. The definition of transfer func­ tions is extended and generalized for synthesis and optimization problems.

2.1 The FEM in the design process of mechanisms

The design philosophy divides the process of designing mechanisms into two major parts, dealing with kinematics and dynamics respectively, see fig. 1.1. The rigid body motion problem is to be solved first: the designer tries to obtain a certain type of mechanism and its kinematic dimensions, dependent on a prescribed "rigid" motion. Next the applied forces are taken into account to determine the link dimensions with respect to strength, stiffness, vibrations etc.

Historically the FEM has been developed to operate in the second part of the design process (statics, dynamics). Its name was taken from the typical aspect of dividing a complex machine part into a great number of simple parts or elements. This allows the calculation of for instance the deforma­ tions of that complex machine part caused by the applied forces.

Application of the FEM to kinematics however requires that forces are left out, and consequently that deformations must be prescribed: usually zero, but occasionally a finite value. The word "deformation" in kinematics refers to "change of form" caused by experiment of mind rather than caused by forces! The kinematic description of a mechanism does therefore not re­ quire links with complex form. Usually it is allowable to simplify:

"Each link is an element", and

"Each deformation concerns one kinematic parameter".

The latter statement plays an important role in kinematic synthesis and optimization. The aim to calculate kinematic parameters must be interpreted as: to calculate deformations, not as a result of forces, but as a result of a certain strategy of prescribed variations of these deformations.

To distinguish between a kinematic parameter and its variation the words "form parameter" c and "deformation" Ac will be used.

2.2 Description of elements: the continuity equations

The kinematic model of a mechanism is defined with elements. Each ele­ ment is assumed to be determined completely by its coordinates in the global

(or fixed) coordinate system xOy. The coordinate values specify the position of that element, while the motion of the coordinates determines the motion of that element.

This will be explained for an example, the binary element of a planar linkage, which is determined by the vector of coordinates x of its end points P and Q (see fig. 2.1). In vector notation:

x = |xp yp xQ yQ|

Each element is defined such that its coordinates are linearly independent. Kinematic relations between the coordinates may however exist by prescribing deformations Ae, which can be expressed as the difference between the actual form parameter c and the prescribed form parameter e :

Ac (2.1) coordinates x = ixp yp xQ yQ |T deformations Ai = L - i° and/or Ap = p - p°

D:x»

>V

+ A

P

i Ay,

A y J > ^ A x

Q P AXp ; U \ A xQ- xp)2 +( yQ- yp)2

DD(x):

Ax-»-A l Ap sin P : cos P y p - y p I XQ ~XP I - c o s P - s i n P cosP sinp sinp -cos|3 -sinp cosP

I I L L

A x

p

*y

p

A* Q

Fig. 2.1 The binary element. Coordinates, deformations and continuity equations D and DD.

-2.3-The actual value E ( X ) is then a function of the coordinates x of only that element.

For the planar binary element frequently the length £ will be pres­

cribed, which defines the element to be "rigid". It is however not a pre­ requisite for an element to be "rigid". If for instance the angle B is prescribed, the element will be kept in a fixed angular orientation. Prescribing both 2P and Bp would define a "rigid" link, which is still able to perform a translatory motion in the xOy-plane. These relations, which are frequently used for this element (see fig. 2.2), can be expressed as

prescribed deformations in accordance with (2.1) as:

elongation A 2 rotation AB

IT , and

„P (2.2)

In (2.1) and (2.2) the deformations express either finite or infinitesimal quantities. The actual length 2 and angle B are functions of the coordinates x, here:

e(x) - V ( x

Q

- V

2 + (y

o "

y

p

) 2 > and

yQ " yP x0 " XP

B(x) from sin B = — - and cos B = -^—

(2.3)

Type of element Deformations Al = 0 bar or truss sliding pair without turning sliding pair linear actuator* rotatory actuator AP=free A U f r e e Ap=0 A l =free AP=free Kinematic drawing A U p r e s c r i b e d JO AP=prescribed JO eliminating a free deformation

c^r

Fig. 2.2 The planar binary element, the type depending on which deformations are free or prescribed.

k k k

In general the (non-linear) function e = e (x ) , with superscript k for element number k, is called: continuity equation of the element k.

To calculate x for given e, later on for the whole mechanism, it is necessary to solve-the system of non-linear continuity equations to be as­ sembled with all the elements of the mechanism. The solution method makes

k k

use of linearization, which requires differentiation of E (x ). Infinitesi­ mal variations of x and E can be related by:

a k

, k i . k _k . k Ae. = Ax . = D. . Ax .

i 3x j x,j J (2.4)

Repeated subscripts, like j in this first order continuity equation, indi­ cate summation (index notation). The comma in D. . indicates differentiation with respect to the coordinates x. For the binary element one can derive:

Ack = A?

AfT sing -CQS& - s i n g cosE

H i 2 2 i xp A yp A XQ i yQ (2.5)

The general first order continuity equation of element k can also be written in matrix notation a s :

Ac = D Ax (2.6)

In kinematic analysis this relation between displacements Ax and deforma-tions A E can be used in three different ways [ 6 ] , see also fig. 2.2:

ko

To prescribe zero deformations: Ae 0. This implies a condition on the position and the motion of the coordinates.

- To prescribe non-zero deformations: Ac * 0. This implies an infinitesi­ mal or finite step of motion. A practical use is to define a degree of freedom of a mechanism.

kr

- To calculate non-prescribed (free) deformations Ac , depending on varia­ tion of coordinates.

The ideas concerning large deformations, which are essential for mechanisms, will be explained in the next chapters. Defining E as the vector space of form parameters of element k, E contains the the direct sum of the three subspaces:

-2.5-Ek = Ek° 9 Ek m 6 Ek r (2.7)

Elements used in this research work are kinematically described in appendix A. The reader will find for each element type the definition of coordinates, deformations and fundamental matrix D . For detailed

types of elements, both planar and spatial, see [10]. k

deformations and fundamental matrix D . For detailed information about other

2.3 Mechanisms and their generalized first order transfer functions

A mechanism can be understood as an assemblage of links (elements). According to the FEM-theory a structure or a mechanism is defined completely by its assembled coordinates. The vector space of coordinates X is the union of coordinate subspaces X of the individual elements:

x 6 X = U Xk (2.8)

The vector space of form parameters E is the direct sum of the subspaces E of the individual elements:

c e E = 9 Ek (2.9)

Contrasting with a statical construction, elements of a mechanism are sup­ posed to make large movements. Therefore it has found to be useful to distinguish three types of displacements [6]:

- Prescribed zero displacements: Ax = 0 . This can be used to define a coordinate non-moving, as required in a fixed pivot.

- Prescribed non-zero displacements: Ax ^ 0 . This implies an infinitesimal or finite step of motion. A practical use is to define a degree of freedom of a mechanism (compare also the possibility concerning a prescribed form parameter).

- Non-prescribed displacements Ax , intended to define free (or dependently moving) coordinates.

The vector spaces of coordinates and form parameters of a mechanism are X and E respectively:

X = X° @ Xm e XC (2.10)

Distinguishing coordinates and deformations is a specific action i n "topolo­ gy definition". T h i s can b e done by designers with elementary u n d e r s t a n d i n g of mechanism theory. S e e f o r instance f i g . 2.3 for a n example "by hand": the p l a n a r four b a r linkage A A B B w i t h coupler point C , defined with only b i ­ nary elements. Note that subspace E , defining the single degree o f freedom of this mechanism, contains the angle (5. , a form p a r a m e t e r o f e l e m e n t 1. T h e subspace X is empty. deformations coordinat Ae = Al, A l2 A p2 A l3 A(33 A l4 A

P

4 A P5 = 0 Ax = = input. = 0 = free = 0 = f ree = free = 0 = free A xA „ AVA0 A xA AyA A*Bo A* B O A xB AyB A xc Ayc es = o = 0 = free = free = 0 = 0 = free =free = free = free

DD: D.Ax = Ac

DD

C

: D

c

.bx

c

= Ae°

Ae = Ae° Aer prescribed free Ax" Axc fixed free

Fig. 2. 3 Mechanism consisting of binary links. Prescribed and free coordinates and deformations.

The continuity e q u a t i o n o f a mechanism'can be expressed with t h e map D:

D: X ♦ E ; e = D ( x ) (2.12)

However, kinematic calculations usually concern the inverse map o f ( 2 . 1 2 ) : the coordinates m u s t b e calculated a s function o f t h e d e g r e e s o f f r e e d o m — s u c h as the prescribed form parameters E . This kinematic function is called "transfer function". B u t either as the ultimate o r as the i n t e r m e d i a t e r e s u l t , the calculation o f this transfer function m u s t b e carried o u t first. F o r the purpose o f k i n e m a t i c optimization the definition o f the w o r d

-2.7"

"Transfer functions concern all,dependent geometric motion quantities (both coordinates and form parameters) and their derivatives as function of all prescribed motion quantities (geometric input)."

„ c c, o m o m, So x = x (c , e , x , x )

, r r, o m 'o m. and c = c (c , c , x , x )

(2.13)

are the basic nonlinear equations to solve in kinematics. The linearization of these equations can be obtained from the first order continuity equation of the mechanism:

Ac = D.Ax (2.14)

or, with the distinctions previously defined:

Ac° Acm Ac1" =

D°° D

0 m

D

O C _mo p. mm _mc Dro Drm Drc Ax° Axm AxC (2.15)

To calculate the transfer functions one may start by writing Ax explicitly using the first two equations of (2.15):

AxC = _Ac Ac D00 Dom Dmo Dmm Ax Ax (2.16) Defining x ,c 3x7 3a. J lim AxC l Aa.-»0 j Aa

k«j=°

with a. as element j of the assembled vector a of prescribed motion quantities:

r o m o m-i

a = [c , e , x , x J 2.17)

the first order transfer function x' (matrix of partial derivatives) can be derived from (2.16), denoting I for the Identity matrix, as:

,-

c , c , c 3x 3x , o . m 3e 3c c c 3x 3x T-l

D°° D °

m Dmo Dmm (2.18)

Without loss of possibilities the concept expressed by (2.13) to (2.18) can be simplified considerably by:

- Omitting the use of prescribed coordinates x to define degrees of freedom of a mechanism. By proper use of elements any degree of freedom can be defined by prescribing form parameters E .

- Omitting the use of variations of fixed coordinates x , as could be desired in kinematic optimization (or even in dynamics). Variations of fixed coordinates can also be obtained by prescribing form parameters E . Unless stated otherwise both assumptions will be maintained in this research work, which permits the definition of the first order transfer function as:

,.c =

3x£ 3x°

, c „ c 3x_ 3x , o , m 3c 3 E (2.19) .,r 3e 3E 3 E , r , r 3 E2 3 E2 3E = D re ,c (2.20)

The matrix will also be noted as D - matrix. In accordance with

(2.19) the inverse matrix contains the first order transfer functions con-cerning all moving coordinates x , derived with respect to all prescribed form parameters e (both c and E ).

All other calculations, either in kinematics or in dynamics, start with this matrix and its inverse. The matrix D must therefore be regular, which is in general true for feasible mechanisms. Unless stated otherwise in this report it will be assumed that:

- D is a square matrix (the same dimensions for the subspaces X and E ) . - The prescribed form parameters e are linearly independent.

The regularity aspect emerges also in the integration procedure to calculate the transfer function of order zero, which is described in the next chapter. For a better understanding the reader is encouraged to study the examples of the D - matrices described in appendix A.

-2.9-2.4 Iterative calculation of the zero order transfer function

The iterative scheme to calculate a sequence of mechanism positions for prescribed zero deformations Ac = 0 and finite deformations Ac is described in fig. 2.4.

As the vector of coordinates x defines the position of the mechanism completely, the aim is to calculate the sequence of x-vectors. This sequence is the numerical result of calculating the zero order transfer function of coordinates. As the fixed coordinates x are assumed to have constant values, the iteration scheme concerns in the first place the dependently moving coordinates x .

To define the starting position of the mechanism it is required to spe­ cify initial values for the coordinates x. In kinematic analysis however usually the form parameters e and e are given quantities. Now it must be decided whether the form parameters dominate the coordinates x or not in the starting position. The scheme assumes all form parameters dominating, so the initial values of the dependent coordinates approximate the starting position of the mechanism.

The main loop provides firstly the calculation of all actual form para­ meters c(x), matrix D and its inverse. Dependent on sufficiently small de­ viations Ac and Ac either the "correction branch" is entered to improve the current mechanism position or the "prediction branch" is entered for a first estimation of the new mechanism position. The correction action is taken by prescribing the opposite actual deformations (deviations). The pre­ diction action is prepared by updating c with the desired finite step Ac ,

o c while all Ac remain unchanged. By Taylor expansion the coordinate vector x

can either be improved or predicted for the new position. c

This scheme holds for any mechanism with regular matrix D , which means for any regular mechanism. Singularity problems may occur with respect to limits for a certain degree of freedom c [12, 13]. For optimization purpose the singularity problem must however be extended to all form parameters, which leads then to the limits of the kinematic parameter space as discussed in chapter 4.

It can be concluded that the zero order transfer function x can be cal­ culated using its first derivative x' and the zero order continuity equa­ tions concerning prescribed form parameters c and c . These calculations

r r r c are always required, all other calculations like c , c' , c" and x" are

optional . Optional calculations may take place in the prediction branch afterwards, that means after having reached the correct position. They are established further in chapters 2.5 and

3-c

START

3

mechanism, starting position Topology Geometry x = Parameters E Degrees of freedom e calculate e (x), e (x) c c — 1 calculate D and [D ] AE = c - c . m m Ae = e - E

-(AC°

deviations from prescribed values E , E in current position

and AE small enough?

D

yes

other calculations: x" . e , e ' , e"

(

optional

new position required?

>

yes current position update (corr) . o , o Ac ■> - A E , m . m Ac ■» - A E new position f READY \ update: m m t m E ■» E + A E

update degrees of freedom (finite step) extrapolation: c c x ■» x _{[ D0] "1.} AE Ae extrapolation x , either "prediction" (new pos.) or "correction" (current

position)

fig. 2.4 Iterative computation scheme for the zero order transfer function x and other transfer functions.

-2.11-At this place it is remarked that the double geometry description in the starting position (both coordinates and form parameters) avoids the problem of dual representation of a mechanism. In the example of fig. 2.5 the points B or B respectively determine uniquely which of the two dual four-bar linkages is to be considered!

A

o

A B B

0

and A

0

A B B

0

are dual mechanisms

Fig. 2.5 Dual four-bar linkages.

2.5 Calculation of the second order and other transfer functions

The second order transfer function of coordinates x" can be derived from (2.19). For this purpose index notation is preferable to matrix notation. With 6., as the

ik

mula (2.19) is written as:

notation. With 6., as the Kronecker delta (compare the Identity matrix)

for-Dc . . x'.c. = 6.. (2.21)

i.J J.k ik

Differentiation of (2.21) with respect to any e leads to:

. . x" , + D. x' , x' = 0 i . J J . k l i , n m n , k m , l

and then, substituting (2.19) again, to:

x':c. , = - x'.c. Dc x'c. x'cn (2.22)

3 x .

-t C

which is the "matrix block" " — with all second order derivatives x to

c i e . The block D. , containing all partial derivatives , must be

i,nm „ c, c _{3x 3x}

n m

assembled with the second order matrices of the elements in accordance with the rules (2.8) through (2.11). For a detailed description of these second order matrices of the elements see appendix A.

Although the transfer functions of coordinates have been derived now for order 0, 1 and 2, which define thereby the complete motion up to order two. of all elements, it is often useful to calculate other motion quantities. The transfer functions of dependent form parameters can be considered here, because they can be described in a general way.

The zero order transfer function e (x) is supposed to be available for each element, see (2.3). The first order transfer function E ' has been de-rived previously: (2.20) calculates the assembled matrix c' of the whole mechanism. It also possible to calculate a specific column vector of this matrix. Let e concern a form parameter of mechanism element k, then

e.k = Dk x,k ( 2 2 3 )

K

will calculate its derivative values dependent on (row) vector D of the first order continuity matrix and matrix x' of the first order transfer function of that element k.

To derive the second order transfer function, which can be done for each form parameter individually, (2.23) i s rewritten in index notation:

C',ki = ^,1 X'l,i ( 2-2 4 )

Differentiation yields:

e"k. = Dk x:,k.. + Dk, x'k. x'k. (2.25)

>ij ,1 l.ij ,1m 1,1 m,j

32£k k

which is the matrix of all second order derivatives of c with res-3cP3eP

l J

pect to e . Because all prescribed form parameters ep are linearly

independ-ent, this will turn out to be a zero matrix for e being a prescribed form parameter!

Note that the transfer functions, as described above, concern functions of form parameters e and e only. So far no distinction has been made

be-

-2.13-tween E and e , so their role may be interchanged, as will be done occasionally in the chapters later on concerning optimization theory.

2.6 Summary and conclusions concerning the FEM-approach for kinematic analysis

It can be summarized that kinematic motion is defined by the generalized transfer functions, which concern the coordinates of the elements constitut­ ing the mechanism. In kinematic analysis problems they depend, for a given mechanism, on the degrees of freedom only. For kinematic synthesis problems, including optimization, the dependency must be extended to all prescribed form parameters.

Q

Non-trivial mechanisms have the same number n of prescribed form para­ meters (including the degrees of freedom) and dependently moving coordina­ tes. The generalized transfer function of order zero is a vector of single transfer functions, the first order transfer function contains a matrix and the second order transfer function a block etc. The dimensions are n ,

c 2 c 3

(n ) and (n ) respectively if only the dependently moving coordinates are considered.

The FEM approach requires an iterative calculation procedure for the transfer function of order zero by prediction and correction. This procedure makes use of the continuity equations (order zero and one) concerning all mechanism elements. An initial guess of coordinate values is required to start the calculations. To displace the mechanism to a new position the pre­ vious position is always utilized: the mechanism is said to "move" from position to position.

For various types of elements, either in planar or in spatial descrip­ tion, the continuity equations have been defined and incorporated in the appropriate computer programs [10, l*t, 15]. Using these programs the mecha­ nism is to be specified by a list of elements, linked together according to their numbering of coordinates (topology table), which is said to be done "at input level". Besides the ease of use, since no formulas are needed from the user, the big advantage is the availability of all first and second or­ der transfer functions, which are automatically calculated in each mechanism position. The current set of elements is sufficient to describe all mecha­ nisms . Nevertheless there might occasionally be a need for additional ele­ ments, which are supposed to be more efficient in a particular case. For such cases the programs [14, 15] allow user-defined elements.

The transfer functions of coordinates define the basic motion quantities to be calculated in kinematics. In the same generalized way a second type of

transfer functions is defined: those concerning dependently moving (non-prescribed) form parameters. As the motion of each element is completely defined by the motion of its coordinates, these transfer functions can be calculated afterwards individually, as far as desired in the kinematic problem. The same is true for all other motion quantities, including time dependent ones. A more intensive description of relevant kinematic motion quantities, which may play a role in kinematic synthesis and optimization, is treated in the next chapter.

-3-1-3. KINEMATIC MOTION QUANTITIES DEFINED BY TRANSFER FUNCTIONS

In practice designers of mechanisms use many types of motion quantities to specify their motion problem. Such quantities can often be identified as one of the transfer functions previously defined, or they can be built using the transfer functions. This chapter describes those kinematic motion quantities which are frequently used in kinematic syn­ thesis and optimization. Prescribing a motion quantity implies a condi­ tion on the kinematic parameters. Referring to (1.3) and (1.4) differen­ tiation of the motion quantities, with respect to the kinematic para­ meters, is considered.

3.1 Single transfer function values and motion quantities in one mechanism position

The motion problems such as those described in fig. 1.2 concern motion c r

quantities of the zero order transfer functions x and E only. In fig. 1.3 the single first order transfer function (J' (a) is the subject of a motion problem for rocker angle (J as function of crank angle a. According to the FEM-approach this motion quantity $' can easily be recognized as a

coeffi-cient of the matrix e' , which is the generalized first order transfer func­ tion of form parameters (2.20, 2.24). Differentiation of such single trans­ fer function values with respect to kinematic parameters like the form parameters, is not at all a problem: they are part of the (generalized) transfer function one order higher (2.25). Up to order two, they are described in the previous chapter.

Theoretically any higher order transfer function could be derived, but for practical reasons there is of course a limit. The expression for the third order transfer function can easily be obtained by differentiation of (2.22). The complete calculation, as necessary for synthesis problems of

c c 4 mechanisms concerning n dependently moving coordinates, would contain (n ) single functions. For example: for a "medium size" mechanism wi th n = 20 there are 160,000 third order transfer functions! To perform these calcula­ tions the third order continuity equations must be available for all ele­ ments, and in the present state of FEM this is unfortunately not the case. This is another barrier to get started with third order motion. Third order transfer functions do not (yet) find application in the practice of

mecha-nism synthesis and optimization. Apparently they are however needed "inter nally" in kinematic optimization,., if second order motion problems are to be solved with any gradient method. In the present state of the FEM-art the only way to obtain third order motion quantities seems to be: numerical differentiation. The consequences are discussed further in the chapters con­ cerning optimization techniques (chapters 5-2 and

5-3)-In appendix B a list of motion quantities and their derivations is pre­ sented, which can be specified in a specific position of a planar mechanism. Except for the transfer functions themselves they concern for instance:

- the position of a point (2 coordinate values),

- the first order motion of a point, which can be described either by the first derivative values of the two coordinates or by the "tangent direc­ tion" and the geometric equivalent for "velocity".

3.2 Motion quantities concerning more than one mechanism position

Motion quantities can be prescribed to mechanisms in arbitrary combina­ tions. It is possible to consider more than one motion quantity in a certain mechanism position. For instance: the position of a point (two conditions) and its motion direction (one condition). Some combinations concern more than one mechanism position. The "stroke" of a motion for instance, defined as the difference between the maximum- and minimum value of any zero order motion quantity, concerns two mechanism positions. Fig. 1.3 shows an example of such a motion problem. In the same way, one may consider the "height" or "width" of a planar coupler curve. Such combinations can be very effective in practice to specify many types of motion problems.

3.3 Kinematic motion: time dependent or time independent

The transfer functions concern so called "geometric motion", which means they are functions of geometric input motion quantities. To define time de­ pendent (output) motion, the input motion must be considered as a function of time. This is however only consistent for prescribed form parameters c ,

o c not for E . For dependently moving coordinates x of a mechanism with a cer­

tain number of degrees of freedom E . can be defined:

xC(c°, c"(t)) , i=l,2 (3.1)

-3-3-.c X dxC . d t 3xC " aem ' i j m d e . ï . ;m-■ ~^T~ = x' . . e . d t , i i (3.2) and accelerations: _ ,2 m _ , m , m ,2 c , c d e. .2 c de. de . d x 3x i 3 x i ~ 2 , m , 2 , ra, m dt dt dt 3e. dt 3e.3e. i i J , "m „c .m .m ._ _. = x'. e. + x".. c. c. (3-3) ,i i ,iJ i J

In the same way time derivatives for dependently moving form parameters e , or for any other kinematic motion quantity, can be defined.

Time dependence plays a significant role in dynamic motion, but this is not the scope of this work. In kinematic motion time dependence can be con­ sidered only if the time relation can be described without forces. In prac­ tice this is usually done under certain assumptions like: "constant input motion velocity: e = 0 or ê prescribed", or "start behaviour: ê = 0, e prescribed" etc. Control of input motion, as applied in industrial robots (open or closed loop multi degree of freedom mechanisms), is certainly a natural application to describe the output motion as time dependent. In kinematics however, the assumption of ideal control of position, velocity, and/or acceleration of input motion is required.

A special category of multi degree of freedom mechanisms exists which can adequately be considered without use of time. It concerns mechanisms with adjustable links which vary either slowly or while the mechanism is not working. See fig. 3-1 as a fictitious example of a two degrees of freedom mechanism with an adjustable stroke in a combustion engine. Stroke s -s .

max min adjustment of the combustion engine is achieved by adjustability of e. , the second degree of freedom. Such mechanisms have a certain prevailing degree of freedom e., the other degrees of freedom (for adjustability) can be rela­ ted to the prevailing one. Comparable with (3-1) through (3-3) differentia­ tion with respect to input motion quantities e yields:

xC(e°, 6™{e»)). i=l,2 (3.4)

j c , c d e .

,c dx 3x i_ ,c ,m

x = = . = x' . . e (3 • 5)

, m m m ,i I \J ->/

de. 3e. de. I l l

A2 C d x ,, m. 2 d ^ ) ,2 m' , c d e . ix ï _ , m , m ,2c de. de . 3 x i j , m -, m, 2 ^± d(c1) „ m , m , m , ra 3 E .3c . de. de, i j 1 1 ,C „m x . e. .1 1 „c ,m ,m x" . . e '. e '. .ij i J (3-6)

In case that no adjustment action-is taken, the adjustable input can be expressed a s :

e. = input motion , e'

m, m, . ,m e.(e.) = constant ,e.

l 1 l

,cf = 0

,e'.' 0 for i*l (3-7)

which reduces ( 3 - 4 ) , (3-5) and (3-6) to the transfer functions of a single degree of freedom mechanism.

In kinematic analysis time dependence has always an assumptive charac­ ter. In kinematic synthesis (optimization) the time relation should essen­ tially be considered as calculable. This is however only non-trivial if the input motions of the mechanism are meant to b e time controlled according to the calculated time relation. The consequence i s , that the motion quantities describing the time relations must be understood as "kinematic parameter". Because of the application in multi-degree of freedom mechanisms this sub­ ject will be discussed in the next chapter.

1 ji?

_S

™>x2

sm a x 1 a ■ \ strc}ke2 strbkel B=CC comb. 1 2 3 A m E1 max m i n max m i n e2 p o s l p o s l pos2 p o s 2 prescribed output m o t i o n quantities 5max1 5m i n 1 smax2 sm i n 2 s ' = 0 S ' = 0 S ' = 0 s ' = 0 E j adjustable comb, 1 2 3 A prescri yes no yes no Ding in _m e2 } n o } no Dut mot •m Ei 1 1 1 1 on quant. _'m e2 0 0 0 0

Fig. 3.1 Two degrees of freedom mechanism, adjustable stroke in a combustion engine.

-3.5-3.4 Motion quantities concerning multi degree of freedom mechanisms

The ideas of time dependence will be illustrated with an example: the five b a r linkage of fig. 3-2, which has two degrees of freedom e. and e_. Consider for instance the motion quantity "direction of motion" ♦ of point C. It will be clear that, corresponding with the definition of the transfer functions, two directions can be defined: one for e. and one for e_ (see also appendix B ) . The "final" direction depends also on the time. Using

(3-2) it follows: tan <t> tangent of curve m-m velocities sA ,sB ,sc angular velocities è™ è™ +J ( f i r s t order input motion quantities)

(3-8)

l|s

A

ll = a.|êTl

lsB»=b.|è^|

Fig. 3.2 Two degrees of freedom mechanism. Assumed time controlled input motion.

In addition to (3-1), zero order motion quantities M (first order M and second order M ) of multi degree of freedom mechanisms can generally be ex­ pressed as: M(c°. e»(t)) M / ° mi^\ .in,. , > 3M .m M(c . c±(t), c (t)) = — z 3e . 1

(3-9)

(3-10)