This document discusses a method for simulating interactions between objects in mechanisms using small-scale interference detection. It represents the contour shapes of objects using molecular models, where molecules prevent overlap between driver and driven objects. It can analyze and synthesize planar kinematic pairs. Examples show it can simulate gears, cams, ratchets, and generateva mechanisms. The method is generalized and automatically optimizes shapes based on functionality plots and molecular time plots. Local shape changes are shown to affect the simulation results.

1. KINEMATIC ANALYSIS AND SYNTHESIS OF MECHANISMS
VIA SMALL-SCALE INTERFERENCE DETECTION
RakeshGupta and MarkJ. Jakiela
ComputerAided Design Laboratory, Massachusetts Institute of Technology
Departmentof MechanicalEngineering, Cambridge, Massachusetts 02139
Abstract
Amethodto simulate the interactions betweenobjects of a
mechanismis described. The methodcan simulate planar
kinematic chains of any length. The samemethodis used
with modifications to generate the geometryof one object
of a kinematic pair given the required functional
relationship and the geometry of the other object. The
planar contour shape of each object of a kinematic pair is
represented as an arrangementof very small entities called
molecules. Given the motion of the driver, the motion of
the other object of the pair, knownas the driven, is
determined by global constraints and the prevention of
geometric interference among the molecules of both
members.Themolecularrepresentation allows very detailed
changes in the shapes of the members.This facilitates
automatic optimal shape design and very generalized
kinematic analysis and synthesis. A computer
implementation, intended to be used in the future with
higher-level intelligent processing, is described and
examplesare provided.
Introduction
Recent research in Qualitative Kinematicshas focused on
the issues of symbolically interpreting the geometryof
mechanisms to determine their behavior 0oskowicz and
Addanki 1988, Joskowicz 1989, Forbus et al. 1987,
Faltings 1987, Faltings et al. 1989). Most of these
methodshave beenlimited to analysis of objects for which
the shape can be represented by a combinationof regular
shapes like lines and arcs. Configuration spaces (Lozano-
Perez 1983) have been used as a basis for analysis of
behavior of the mechanisms,using heuristics to establish
the relationship of output parameterto input parameter by
searching through the large solution space of configuration
space plots. Theissue of directly correlating the shapes of
the objects with functional performancehas, on the other
hand, been somewhatneglected.
Qualitative Kinematicsis the aspect of spatial reasoning
concernedwith geometry. Givena set of objects forming a
mechanism, it is necessary to reason whether motion
transmission will be continuous or discontinuous or
whetherthere will be no motiontransfer becauseof locking
of the mechanism. A mechanism is a kinematic chain
consisting of one or morekinematic pairs for transmitting
forces and motion. AKinematic pair is a pair of objects
linked together so that their relative motionis constrained.
ConfigurationSpaceis the set of free placementsof objects
of a kinematic pair so that they do not interfere (Lozano-
Perez (1983)). Aconfiguration space gives all possible
relative placementsof the objects and these mayor maynot
be achievable for a given initial condition of the
mechanism. Wedefine the term functionality as the
relationship of output parameterto input parameter.
Theresearchdescribedin this article is part of a larger effort
to achieve computer-basedconceptual design in mechanical
design domains. Anexamplemight be an Idea Editor where
the designer can sit at a graphics terminal, sketch new
designs, and have the system simulate them without user
intervention to determine the functionality. Sucha system
wouldreplace the traditional sketching using pencil and
paper and spare the designer of the task of visualizing the
complexinteractions of objects.
To comply with a set of design requirements, new or
modified shapes need to be considered. It is desirable to
automatethis process whichis presently done manuallyby
humandesigners. In this article, weaddress this issue by
describing how our system automatically synthesizes
shapesthat wouldbe required to satisfy required functional
performance.
Wehavefocused our efforts on interacting planar kinematic
pairs without friction such as meshing gears, cams and
followers, ratchets and pawls and geneva mechanisms.We
will first introduce our molecularmodelof kinematicplanar
contour shapes. Examples will be discussed on howthe
system can be utilized for analyzing designs and
synthesizing them. Analgorithms for synthesis of designs
will be explained. Wewill then demonstrate howlocal
shape changes affect the simulation by means of an
example. Useof MoleculeTimeplots in shape design will
also be discussed.
Molecular Models of Kinematic Pairs
Werepresent the contour of an object of a kinematic pair
by an arrangementof entities that wecall molecules. A40-
tooth gear, requires on the order of 800moleculesto define
Figure la: MolecularModelof GearMechanism
50
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.

2. Figurelb: CloseupofMolecularmodelofthegears
its shape.Thelowerlimit to this numberis achievedwhen
the smallestcontinuouscurvemakingthe contour(top land
of gearteethin caseof gears)is representedbyat least three
molecules.Anynumbergreater than this will allow the
representation of a finer level of detail but wouldbe
computationallymoreexpensive. Thegear teeth faces in
figure la, for example, are defined by arranging the
moleculeson true involute curves. Themolecules are
shownas small circles in the closeup figure lb. The
molecules that makeup a contour shape are numbered,
normallyanticlockwise. This numberingis used in the
analysisof the kinematicinteraction.
Thesimulationof the interaction betweentwokinematic
objects is baseduponthe simpleidea that twomolecules
cannot be in the sameplace at the sametime. If an
anticlockwiserotationis impartedto the driver gear(onthe
right), then someof the driver moleculeswill beginto
interfere withsomeof the drivenmolecules.Toperformthe
simulation, the driven gear contour mustbe movedin a
waythat is consistentwithits globalconstraints(e.g. that
it mustrotate aboutits center)andthat interferencebetween
the drivenanddrivermoleculesis prevented.
Above, we had assumed the satisfaction of global
constraints. In the systemas implementedso far, global
constraints are not implementedwiththe molecularmodel.
Thegears shownin figure la, for example,do not rotate
abouttheir centers becausea jourual/shaft interaction is
modeledwith molecules. The allowable motion of the
contourmoleculesis explicitly codedinto the softwarethat
runsthe gearsimulation.It is however,possibleto extend
this workandexplicitly modelanentire kinematicsituation
withmolecules,onlyallowingthe externalconstraint that
somemoleculesare boundmotionlessto ground.
Regionsof interactingcontoursthat arenearinterferenceare
calledhot spots,andare identifiedbythe moleculesthat are
hot. Wecan makea plot of hot moleculesversus time for
eachmemberof an interacting pair for a givensimulation.
Sucha plot for the driver gear of figure la is shownin
figure lc. These plots are a wayof abstracting the
kinematicinteraction andcontain a great deal of useful
information.Avertical line at anytimeinstant showsthe
extents of contact regionsandalso indicates if multiple
contact is occurring.Forthe gears it is evidentthat for
50 100 150 200 250 300 350 400 450 500
TIMB
Figurelc: MoleculeTimeplotfordrivingsmallergear
-1.5
-Z5
-3
ANTICL~ ROTATIO~ OF DRIVING GEAR (mc~m)
Figureld: Functionalityplotfor gearmechanism
mostof the time the gears havea contact ratio of two.
Repetitionof clusters signifies contact amongsuccessive
teeth. Thehorizontal gapsbetweenclusters correspondto
moleculeson the addendumand dedendumcircles which
never becomehot. The discontinuous jumpfromlow to
high moleculenumbersaroundtimestep 450in figure lc
indicates repetition of the samecycle. It is evidentthat
theseplots showbothlocal andglobalcharacteristicsof the
kinematic interaction. Weplan to use these plots to
determineappropriatechangesto the contourshapes.Figure
ld is a plot of theinputrotationandoutputrotation.
f
Figure2a: Molecularmodelof Genevamechanism
5]
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.

3. Examples1 through3 serve millustrate the generality of
the simulationapproach.Figure2a showsa genevaandpin
molecularmodel,figure 2b showsthe M-Tplot for the pin
and figure 2c showsthe M-Tplot for the geneva wheel.
Thecharacteristic humpshapes in figure 2c are causedby
contactof oneface of the slot withthe pin. If the pin were
slightly larger, it wouldcontactbothfaces of the slot and
there would be two symmetric humpsin the M-Tplot.
Figure2dshowsthe functionality plot.
5O
40
3O
m
~--=_.
i / i i i
TIME
Figure2b:MoleculeTimePlotfor drivingpin
6OO
50O
!:f
2OO
SO
0
16OO
Figure2c: MoleculeTimePlotfor thegenevaslottedwheel
0,i,
-O.2
1 2 3 4 5 6 7
Figure2d:Functionalityplot for thegenevamechanism
Figure 3a shows a molecular model of ratchet and pawl
mechanism.In this case the systemcan reason that if the
ratchet wheelis rotated clockwise, interference cannotbe
removedbetweenthe objects andthe kinematic pair will
not work.Foranticlockwiserotation of the ratchet wheel,
figure 3b showsthe M-Tplot for the ratchet wheel, and
figure 3c showsthe M-Tplot for the pawl. Wenote that
the simulations for figure 3 required the encodingof a
positional restoringfunctionfor the pawl. Springforces or
inertial effects suchas the reactions to gravity, are not
accounted for in the strictly kinematic simulation’.
Functionalityplot is shownin figure 3d.
It is important to note that the simulations shownin
figures 1 through 3 were performed with a common
algorithm, thus demonstrating the generality of the
approach.Thesystemdoes not knowthat it is simulatinga
particularkinematicpair: all it knowsis the contoursof the
objects. Thesimulationprocedure(GuptaandJakiela 1992)
involves determinationof hot spots andnewposition of the
drivenmembercorrespondingto the driver motionfor each
timestep of a simulation.
Figure3a: MolecularModelfor RatchetandPawl
mechanism
400
3OO h
2O0
150
100
50
0
5’o 1~o 1;o ~o ~o 3~ 35o ~o ,~o
TIME
Figure3b:MoleculeTimeplot for ratchetwheel
50O
1 Wenotethat GardinandMeltzer(1989)haveencounteredthe
sameproblemwith their molecularmodelsof strings and
fluids.
52
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.

4. I0¢
90
80
7O
6O
50
40
301~ ~.,...,.~~.~.k_,.,.~ ~ ~.~ ~ ~ ,.~ k_~.
2°Iio -""-’"---""
°o :’o 1~o 1;o ~ z;o ~o 3;0
TIME
Figure3e: MoleculeTimeplot for pawl
5OO
1
|"t~6
0
-O.6
-O.8
,<
-I i t i
ANTICI.DCKW~BROTATIONOF RATC~RT~I~L (tldiaal)
Figure3d:Functionalityplotfor theratchetandpawl
mechanism
Synthesis Algorithm
Synthesis is the process of determiningthe geometryof the
mechanismto accomplisha desiredfunctional relationship.
Themethodof synthesizing designs is based on material
removal from the largest possible size of a blank
(analogous to hobbing of gears). A flowchart of the
synthesis procedureis shownin figure 4.
A default circular contour is madefor the object to be
generated, with radius (c-x). Here x is the smallest radial
distance of any molecule on the given geometric object
fromits center of rotation and c is the distance betweenthe
centers of the twoobjects. Therequired functionality data is
then split up into small steps such that a time step
transition will not moveany moleculeby morethan half a
molecule diameter. The center distance between the two
objects is increased by the amountof overlap betweenthe
twoobjects, and extra time steps are addedto the beginning
of the required functionality data so that the initial center
distance configuration is an intermediate translational step
of the synthesis process. This is necessary becauseif the
additional timesteps are not added,the initial configuration
of objects mayinvolve someoverlap of the two objects
Addand delete
molecules
Advancethedriveranddsive~globally[ =
accordingto thegivenrequiredfunctional
linput-outputrelationship
Makea listofclosepairsofmoleculesI
forthetwointeractingobjects
I
f
Movethe driveamembermoleculesI
locallyina directionawayfromthepaired|
drivermoleculeonthelineofcontact[
!
intersecdon in the
member being
genesrted
~~__[Evenly space the molecules ]
on the new member I
I
Figure4: FlowChartfor the synthesisprocedure
with part of one object inside another, whichis not a valid
mechanism.
Thedriver and driven objects are movedglobally according
to the required functional input-output relationship. Alist
of close pairs of interacting moleculesis madefor the two
interacting objects. Thehot driven molecules are moved
locally in a direction awayfromthe paired driver molecule
on the line of contact, by a tenth of molecular diameter.
The distances between consecutive molecules on the
generated object are checked and if the distance becomes
morethan a moleculediameter, a newmoleculeis added. In
case molecules becometoo close molecules can similarly
be deleted. This is repeated until the interference has been
removed.
Thegeneratedprofile is then checkedfor self intersection. If
the contourself intersects, it is split into twoparts. Atthe
end of the cycle, the moleculeson the generated memberare
equally spaced, and the mechanismso obtained can be
simulatedas discussed in the earlier section to verify that
the functionality is the sameas the one that wasused for
its generation.
Figure 5a showsthe initial setup for generation of a geneva
slot, given the required functionality as that for a geneva
mechanismand the geometryof the pin. Starting with the
geometrygiven in figure 5a, the synthesis producesa slot
as shownin figure 5b. TheV-shaped chamfer at the open
end of the slot has been obtained due to molecules being
removedfrom the area by the pin. If westart with a lower
diameterof the blank, weend up witha parallel slot.
53
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.

5. Figure5a:Givenpin andstarting blankfor thesynthesisof
a genevamechanism
Figure5b: Theslot asgeneratedbysynthesisprocedure
Figure6a: Givenslotted wheelandstarting blankfor pin
synthesis,positionedat correctcenterdistance
Figure6b: Thegeneratedcontourshowingself intersection.
If the geometryof the slot is given, with the required
functionality of a geneva mechanismthe initial center
distance is reachedafter sometranslation steps in synthesis
(figure 6a). Apart of the pin contour gets separated from
the blankdue to self intersection (transition fromfigure 6b
Figure6c: Thegeneratedcontourseparatedin twoparts
after self intersection
Figure6d: Thepin as generatedby synthesisprocedure
to 6c). Thepin is not exactly circular due to numerical
approximationof its shape by moleculesof finite diameter.
Anotherreason is that all the moleculeson the pin do not
comein contact with the slot. As the simulation goes on
in time wefind molecules being deleted from the contour.
Asthe simulation nears the end the contour breaks up into
smaller parts due to self intersection, which finally
disappeardueto interference (figure 6d).
Effect of Local Shape Changes on M-T Plots
This exampleis included to demonstratethe sensitivity of
the molecular time plots and to indicate howthe system
could be used for the optimization of kinematic shape.
Figures 7 and 8 show close ups and M-Tplots for tooth
faces with no curvature and tooth faces with convex
curvaturerespectively. Figure 8 is quite similar to, but not
exactly, a true involute.
Figure7a: Closeupof molecularmodelof line teeth gears
54
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.

6. -- °
i
30O -~
]
200 -
- _~..._
lot -
- -....
FigureTo: MoleculeTimePlot for thedriving gearw~hline
teeth
Figure8a: Closeupof molecularmodelof convexteeth
gears
500
4oo
2ot
-......._
--..,....
-,,....,,._
tot -~.~
...,...
i i i i i i i
SO 100 I~1 200 250 ~ 350 400 450 ~10
TIME
Figure8b: MoleculeTimePlot for the driving gear with
convexteeth
M-Tplots can indicate that certain regions of the tooth
contour contact disconnect, and contact again for the
straight and concavegears. Suchan effect is very hard to
predict beforehand and is evident when watching the
simulation animation. Surfaces such as tooth faces could be
modified and simulated in an iterative mannerin order to
find an opdmalface shape. Weare investigating the use of
binary imageanalysis algorithms for this purpose.
Discussion
The interference check of the simulation is quite
quantitative, with no qualitative (high-level) reasoningdone
to perform the simulation, Theresults of the simulation,
however,produce the qnalitative behavior of the kinematic
pairs and the representation of the results, in the form of
molecular-time plots and functional plots, will allow
qualitative reasoning on these results. In particular, we
believe that our approach will produce consistent
correspondencesbetweenqualitative behavior and physical
form.
Thesystemis written in "C" and runs on Silicon Graphics
Iris 4D/’70-GTgraphics workstation. Atypical simulation
of a kinematicpair with all computationsin real time takes
approximately 5 minutes. A typical synthesis of a
kinematic pair for one rotation of one of the objects with
all computationsin real time takes 10-15minutes.
Acknowledgments
This workis supported by the NSFwith a Presidential
YoungInvestigator’s grant (DDM-9058415).Matchable
funds for this grant have been provided by the Ford Motor
CompanyFundand by Polaroid Corporation. Additionally,
the systemwasdevelopedusing the computationalfacilities
of the Computer-Aided Design Laboratory at the
Massachusetts Institute of Technology, Department of
Mechanical Engineering. These sources of support are
gratefully acknowledged.
References
Faltings, B., 1987, "Qualitative Kinematics in
Mechanisms,"Artificial Intelligence, Vol44, July, pps.
89 - 119.
Faltings, B., Baechler, E., Primus, J., 1989, "Reasoning
About Kinematic Topology," Proceedings of the
International Joint Conferenceon Artificial Intelligence,
Detroit Michigan,laPS 1331- 1336.
Forbus, K. D., Nielsen, P., Faltings, B., 1987,
"Qualitative Kinematics: A Framework,"Proceedings of
the International Joint Conference on Artificial
Intelligence, Milan, pps. 430- 435.
Gardin, F., Meltzer, B., 1989, "Analogical Representations
of NaivePhysics," Artificial Intelligence, Vol. 38, No2,
pps. 139- 159.
Gupta, R., Jakiela, M. J., 1992, Qualitative Simulation of
KinematicPairs via Small-Scale Interference Detection,
ASMEFourth Int. Conference on Design Theory and
Methodology, September, Scottsdale Arizona.
Joskowicz, L., Addanki, S., 1988, "FromKinematics to
Shape: AnApproachto Innovative Design", Proceedings
of the AmericanAssociation for Artificial Intelligence
Conference, Saint Paul Minnesota, pps. 347 - 352.
Joskowicz, L., 1989, "Simplification and Abstraction of
KinematicBehaviors," Proceedingsof the International
Joint Conference on Artificial Intelligence, Detroit
Michigan, pps. 1337- 1342.
Lozano-Perez, T., 1983, "Spatial Planning: A
Configuration Space Approach", IEEE Trans. on
Computers, C 32, No 2, pps. 108- 120.
55
From: AAAI Technical Report FS-92-03. Copyright © 1992, AAAI (www.aaai.org). All rights reserved.