Simplification and Abstraction of Kinematic Behaviors - CiteSeerX

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ing the possible motions of objects. Three kinematic constraints are used to model these dynamic constraints: (1) a constant contact relation between two objects ...
Simplification and Abstraction of Kinematic Behaviors Leo Joskowicz I B M T. J. Wat son Research Center P.O. Box 704 Yorktown Heights, NY 10598 E-mail: [email protected] Abstract Efficient p r o b l e m - s o l v i n g in any physical dom a i n requires: (1) the a b i l i t y t o ignore i r r e l evant i n f o r m a t i o n b y i n c o r p o r a t i n g constraints a n d assumptions ( s i m p l i f i c a t i o n ) , and (2) the a b i l i t y to ignore d e t a i l by reasoning at differe n t levels of r e s o l u t i o n ( a b s t r a c t i o n ) . E x i s t i n g analysis m e t h o d s for mechanical devices derive q u a l i t a t i v e descriptions of a mechanism's kinem a t i c behavior f r o m the shapes and positions of its p a r t s . A l t h o u g h q u a l i t a t i v e , these descriptions p r o v i d e a single level of abstraction w h i c h is exceedingly complex and detailed to a u t o m a t e c o m m o n analysis and design tasks. T h i s paper presents a set of operators to simp l i f y and abstract k i n e m a t i c descriptions der i v e d f r o m c o n f i g u r a t i o n spaces. T h e descript i o n h i e r a r c h y defined by these operators provides a basis for m a n y reasoning tasks, such as m e c h a n i s m comparison - d e t e r m i n i n g when t w o mechanisms are k i n e m a t i c a l l y equivalent.

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contours). Each pairwise contact defines a half-space bounded by a one-dimensional curve (called a configuration space boundary or CS-boundary for short) t h a t separates free and forbidden placements. T h e intersect i o n of these half spaces defines the components of the configuration space, w h i c h are then p a r t i t i o n e d i n t o regions. T h e regions, defined by monotone CS-boundary segments, reflect the q u a l i t a t i v e l y different behaviors of the pair. There can be as m a n y regions as there are possible contacts; as a result, behavioral description are frequently too detailed.

Introduction

Recent m e t h o d s for the analysis of mechanical devices derive descriptions of a mechanism's k i n e m a t i c behavior f r o m the shape and the i n i t i a l p o s i t i o n of its parts [Faltings, 87a; 87b], [Nielsen, 88a; 88b], [Joskowicz, 87, 88; 89a]. A l t h o u g h q u a l i t a t i v e , these descriptions provide a single level of a b s t r a c t i o n w h i c h is often t o o detailed and c o m p l e x to be of p r a c t i c a l use in a u t o m a t e d design. T h e c o m p l e x i t y of the b e h a v i o r a l descriptions arises f r o m t w o factors: (1) f r o m the (local) k i n e m a t i c pair descriptions, as a result of t h e o b j e c t s ' shape c o m p l e x i t y , and (2) f r o m the overall (global) m e c h a n i s m descriptions, as a result of t h e c o m b i n a t o r i a l c o m p l e x i t y of possible object positions. L o c a l descriptions are derived f r o m two-dimensional c o n f i g u r a t i o n spaces 1 defined by the o b j e c t s ' degrees of f r e e d o m . T h e c o n f i g u r a t i o n space of a pair of objects is c o m p u t e d by a n a l y z i n g all pairwise contacts between object features ( t h e vertices a n d edges of the objects' *The configuration space of a mechanism defines the set of free placements (position and orientations) of objects so that no two objects overlap [Lozano-Perez, 83].

Figure 1: T h e Half-Gear Pair Consider the half-gear pair in F i g . 1, consisting of a 20-teeth gear and a 9-teeth half gear of equal diameter pinned at their centers to a fixed frame. T h e i r configurat i o n space is shown in F i g . 2(a): dark areas correspond to forbidden object positions; solid lines correspond to CS-boundaries. F i g . 2(b) shows a detailed view of the CS-boundaries and the p a r t i t i o n i n t o regions 2 . Region RQ corresponds to positions in w h i c h the t w o gears are disengaged, and thus their r o t a t i o n s are i n dependent; regions r, correspond to positions in w h i c h the t w o gears are meshed. Each r, region is delimited by two CS-boundaries, corresponding to the upper and lower contact between two teeth. W h e n a contact occurs, the r o t a t i n g one gear causes the r o t a t i o n of the other in the opposite d i r e c t i o n ; the r a t i o between b o t h r o t a t i o n s is given by the f u n c t i o n defining the CS-boundary. T h e fragmented nature of the C S - b o u n d a r y reflects the teeth 2

Both figures adapted from [Faltings, 87a]; for clarity, only half of the "strips" are shown in (a).

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Figure 2: T h e Configuration Space of the Half Gear Pair

contact changes, and indicates slight variations in the m o t i o n transmission ratio. T h e small intervals between CS-boundaries in each region corresponds to gear interplay (backlash), Since there are two regions for each pairwise t o o t h contact, there are 20 x 9 x 2 = 360 r, regions; the regions are qualitatively different due to angular offset and interval differences. T h i s kinematic description is clearly too detailed for a C A D system designing a gearbox w i t h many gear pairs. Global behaviors are obtained by composing all local descriptions. In general, composing local descriptions requires algebraic techniques. However, for fixed axes mechanisms, in which objects can only move along axes t h a t are fixed in space, the composition requires a small set of symbolic rules [Joskowicz, 87]. In the worst case, the composition results in the cross-product of all local regions for every pair, producing an overall mechanism description w i t h exponentially many regions. Consider the cylinder lock in Fig. 3, consisting of a cylinder C r o t a t i o n ally mounted on a fixed frame F. Five pairs of pins of different lengths, ( P t , Qi), are mounted inside five aligned cylindrical holes in the cylinder and the frame. T h e pins are kept in contact by springs and can only translate along the axes of the holes; their role is to prevent the r o t a t i o n of the cylinder. W h e n the right key is inserted, the pins are raised so t h a t the top of the lower pins and the b o t t o m of the upper pins coincide exactly w i t h the outer surface of the cylinder ( F i g . 3(b)). W h e n the pins are aligned, the cylinder rotates together w i t h the key, as there are no obstacles to prevent this r o t a t i o n . T h e cylinder then pushes a tumbler (not shown in the figure) t h a t locks the door by preventing it from r o t a t i n g around its axis. T h e analysis identifies three qualitatively different positions of the pins: (1) the upper and lower pins are in contact w i t h the cylinder, preventing it f r o m r o t a t i n g ; (2) the upper and lower pins are in contact w i t h the fixed frame, also preventing the cylinder f r o m r o t a t i n g ; (3) the top of the lower p i n , the b o t t o m of the upper p i n , and the top of the cylinder are aligned; the cylinder is then free to rotate, in which case the pins cannot translate. T h e key/cylinder pair has two characteristic

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positions: (1) the key and the cylinder are not in contact; the rotation of the cylinder is independent from the r o t a t i o n a n d / o r translation of the key; (2) the key is inside the cylinder; the key is free to translate inside the cylinder, b u t the rotations of the key and cylinder are dependent. T h e global description contains many unreachable regions and irrelevant detail. For example, 3 5 = 243 regions correspond to all the possible combinations of pin positions when the key is outside the cylinder. B u t these positions can only be reached when the pins are directly pushed from the outside! Assuming t h a t i n p u t motions are only applied to the key, all these regions are unreachable. Another source of detail comes f r o m the regions created by the pin positions as the key is inserted: the pins follow the upper contour of the key, thereby changing characteristic positions. A l l these distinctions are irrelevant if we are only interested in the behaviors of the key and the cylinder. T w o ideas play a key role in coping w i t h the complexi t y and focus of behavioral descriptions: simplification and abstraction. Simplification incorporates additional information in the f o r m of assumptions and constraints. A b s t r a c t i o n ignores irrelevant detail by defining different levels of resolution. T h i s paper defines a set of simplification and abstraction operators for kinematic descriptions.

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Describing Kinematic Behavior

T h i s section presents the symbolic language used to describe kinematic behaviors. Objects in a fixed axes mechanism can only rotate or translate (or b o t h ) along axes t h a t are fixed in space. We can thus classify all

their possible motions along a fixed axis into five categories: no m o t i o n , r o t a t i o n , translation, independent r o t a t i o n and t r a n s l a t i o n , and coordinated rotation and translation. To each category, we associate a predicate i n d i c a t i n g the t y p e of m o t i o n and the axis along which the motion takes place. To describe the extent of the m o t i o n , we associate motion parameters bound by intervals defining the motion's range. T h e type of motion, together w i t h the m o t i o n parameters and their intervals f o r m a possible motion label which completely describes an object's kinematic behavior. T h e five possible motion labels for object A along axis O are: fixed p . r o t a t i o n p j r a n s l a t i o n i P-cyUnder and pJielical Relationships between object motions, indicating how objects constrain each other's motion through contact, are specified by m o t i o n parameter relations. Mechanism behaviors are described by a set of possible motion labels (one for each object) and the dependencies between their m o t i o n parameters. Such a description is called a possible motions region of a mechanism's behavior 3 . A region diagram describes all the possible kinematic behaviors of a mechanism and the transitions between them. The region diagram is an undirected graph whose nodes represent possible m o t i o n regions and whose edges represent possible transitions between regions. To distinguish between qualitatively different behaviors, all regions in the diagram must be qualitatively different. T w o possible motions regions, Ri and Rj, are qualitatively different iff at least one of the following holds: (1) the m o t i o n type of at least one object is different in Ri and Rj; (2) the m o t i o n parameter intervals defining Ri and Rj cannot be merged i n t o continuous intervals f o r m i n g a new region Rk = RU Rj; (3) motion parameter relations in Ri and Rj are not identical; (4) motion parameter relations in Ri and Rj are identical but at least one of these relations is monotonically increasing in one region and monotonically decreasing in the other. Region diagrams constitute a symbolic, qualitative description of all the possible kinematic behaviors of a mechanism.

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Simplification and Abstraction Operators

We define simplifications and abstractions as operations on kinematic descriptions and their underlying configuration spaces. To be meaningful, these operations should not introduce spurious behaviors or alter the original relations between behaviors; in other words they should be sound and complete. Soundness guarantees that no new behaviors are introduced, whereas completeness guarantees t h a t no possible behaviors were lost. A t h i r d property, compositionality, guarantees t h a t the simplification/abstraction can be applied to the local pairwise descriptions before composing t h e m . T h e following two subsections briefly describe the operators to simplify and abstract kinematic behaviors. For detailed algorithms and complexity analyses, see [Joskowicz, 89b]. 3

These descriptions are called regions because they correspond to regions of the mechanism's configuration space.

3.1

Local Operators

Local operators simplify and abstract kinematic pair descriptions; they are applied to their two-dimensional configuration spaces defined by n monotone CS-boundary segments. Local operators are applied immediately after the configuration space is computed. There are ten local operators, all sound and complete: Kinematic Constraints Simplifications: specify the object that causes the m o t i o n , the type of motion, and the extent of the motion ( I n p u t - P a r t , I n p u t - T y p e , and Input-Range, respectively). These additional constraints, derived from the context in which the pair is operating, rule out possible behaviors t h a t become unreachable as a consequence of this constraining informat i o n . Input-Part transforms every region R into a new set of regions, r;. In the new regions, the motion parameter of the passive object changes only when the active object's motion parameter changes; there is one ri for every region adjacent to R. I n p u t - T y p e reduces the number of pairwise configuration spaces to be analyzed (one for every pair of fixed axes). Input-Range rules out behaviors that are outside the specified motion range by deleting regions outside the range, and restricting those who lie on the boundary. A l l three simplifications can be implemented in O(n). Dynamic Constraints Simplifications: account for the action of gravity, springs, and friction, thereby restricting the possible motions of objects. Three kinematic constraints are used to model these dynamic constraints: (1) a constant contact relation between two objects; (2) a preferred (or default) position of an object when it is not subject to other contact constraints, and; (3) conditions on the motion relation parameters restricting motion transmission. The first two constraints model the effects of springs and gravity. A force applied to an object causes it to remain in contact w i t h its neighboring objects in the direction of the force. Thus, all positions in which the two objects are not in contact can be ruled out. When no contact occurs, the object w i l l move in the direction of the force until it reaches a stable position. To avoid introducing time information, we assume that motions due to forces are infinitely faster than input motions. The t h i r d constraint models friction by restricting motion transmission through contacts whose CS-boundary tangent at the point of contact is smaller than a predefined coefficient //, regardless of the force's magnitude and objects' masses. These constraints rule out possible transitions between regions and reduces region intervals. All three simplifications can be implemented in 0(n). Linearization Abstraction: approximates the exact motion parameter relations w i t h pieccwise linear relations. Every CS-boundary is divided into monotonically continuous segments, which are then replaced by a set of lines. The new linear CS-boundary might intersect other CS-boundaries, thereby producing a topologically inconsistent abstraction. This is avoided by further subdividing the CS-boundary into smaller segments. Assuming that each segment is broken only a constant number of times, linearization can be implemented using an efficient line intersection algorithm in Q(nlogn).

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Qualitative Abstraction: groups contiguous behaviors w i t h similar monotone m o t i o n parameter relations i n t o one behavior. T h i s abstraction hides the contact details t h a t produce fragmented behavioral descriptions. T h e operator merges contiguous monotone CSboundaries and creates qualitative CS-boundaries. These boundaries indicate the qualitative m o t i o n parameter relation (monotonically increasing or decreasing) in the new regions. Merging CS-boundaries also requires testing for intersections, and thus can be implemented in 0(n log n ) . Linearization and Q u a l i t a t i v e abstraction have been used for the design of new object shapes in kinematic pairs [Joskowicz and A d d a n k i , 88]. Gap-Closing Abstraction: transforms behaviors w i t h small backlash i n t o behaviors w i t h o u t backlash, where object motions are t i g h t l y coupled. T h i s operator merges CS-boundaries defining narrow "channels" of free placements reflecting backlash and tolerancing errors ( F i g . 2(b)). T h e small positional variations inside the channels reflect negligible variations t h a t can usually be ignored. T h e operator examines every region for a possible boundary merge, and can be implemented in 0(n). Behavior Parametrizations: these operators are not simplification or abstraction operators per se: they are designed to compare t w o kinematic behaviors and find a common description by parameterizing their possible m o t i o n labels. T h i s operation is useful for detecting behavioral similarities and periodicity. Possible motions labels have three p o t e n t i a l candidates for parameterization: region intervals, axes of motions, and m o t i o n parameter relations. T w o regions can be parameterized in their intervals when: (1) their m o t i o n type is identical, and is defined along the same axis; (2) the relations between their m o t i o n parameters are identical, and (3) their interval regions are different b u t proportionally scaled. Similarly, t w o regions can be parameterized in their axes when the axes of m o t i o n are not required to be identical. Finally, t w o regions can be parameterized in their m o t i o n relations when these relations are l i n early similar scalarly similar have a phase difference or any combination of t h e m . T h i s operation takes constant time when applied after Linearization. Periodicity Abstraction: detects patterns of repetitive behavior to create a common behavioral descript i o n . T h i s description is produced by identifying parametrically isomorphic subgraphs in the region diagram. T w o subgraphs are parametrically isomorphic iff they are topologically equivalent and there is a one-to-one parametric m a t c h i n g between regions. A l t h o u g h graph isomorphism can be tested efficiently - region diagrams are graphs w i t h a planar embedding - subgraph isomorp h i s m , even for planar graphs, is in N P - C o m p l e t e , and is thus computationally expensive. T h e order in which the operators are applied obeys two motivations: (1) discard as soon as possible the behaviors t h a t , due to constraints, are unreachable, and (2) reduce the number of regions in the diagram as quickly as possible. Therefore, simplification operators are applied first, because they incorporate additional constraints.

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T h e n , abstraction operators are applied in increasing order of coarseness: first Linearization, then Qualitative abstraction. Gap-Closing depends on the results of the previous t w o abstractions and is applied repeatedly unt i l no further change occurs. Periodicity abstraction is applied at the very end, when the local region diagram is smallest. As an example, consider the effects of local operators on the half-gear pair's configuration space of F i g . 2. We assume t h a t input motions are only applied to gear B and t h a t the initial placement is = 0. First, I n p u t - P a r t simplification reduces RQ to a set of disjoint two-dimensional strips bounded by the linear relations and in the interval Only two strips, and are reachable f r o m the i n i t i a l placement. As a consequence, only the regions connected to these two strips are reachable in the interval all other regions are unreachable and are thus discarded. Linearization and Q u a l i t a t i v e abstraction merge the remaining CS-boundaries in the interval into two regions, each defined by t w o parallel lines. Gap-Closing merges the parallel lines in these regions as well as the t w o strips in the interval The successive effects of these operators on the configu r a t i o n space are shown in Fig. 4. Periodicity abstraction identifies two parametrically isomorphic subgraphs, ( r o , r i ) and ( r 2 , r 3 ) , commonly described w i t h a parameter i, where i = 0 describes (r0lri) and i = 1 describes (r2,r3):

3.2

Global Operators

Global operators simplify and abstract behavioral descriptions of complete mechanisms; they are applied to the global region diagram, after the local operators. There are six global operators, most of t h e m generalizations of local operators; their complexity is proportional to the number of regions in the global diagram. Only the first one is compositional. Kinematic Constraints Simplification: specify which objects can receive input motions, the types of these motions, and their extent ( I n p u t - P a r t s , Input-Types, and Input-Ranges, respectively). Like local kinematic constraints, they are derived f r o m the context in which the mechanism operates. These operators further discard unreachable regions and restrict possible motions in the remaining regions. They can be applied while composing local region diagrams, thereby r u l i n g out potential regions w i t h o u t c o m p u t i n g t h e m . Relevance-Set Simplification: filters out behavioral distinctions created by uninteresting objects that introduce too much detail. The user indicates the set of relevant objects whose behaviors s/he is interested i n ; the most common relevance set is the i n p u t / o u t p u t parts set. T h i s operator projects each region into a new region f r o m which the possible m o t i o n labels and relations of uninteresting objects are removed. As a consequence, contiguous regions whose difference lay solely in the behavior of uninteresting objects become qualitatively similar and can thus be merged. T h i s operator is both sound and complete, b u t not compositional. Region-Difference Abstraction: defines how qualitative behavioral distinctions are made. By relaxing the criteria t h a t establish the difference between two kinematic behaviors, coarser behavioral descriptions are obtained. We consider t w o relaxations of the qualitative region difference criteria (section 2): (1) m o t i o n parameter relations can be monotonically identical, instead of identical. T w o relations are monotonically identical iff they both specify the same relation ( < , =, or >) and their functions are b o t h simultaneously increasing or decreasing in the given interval, and (2) M o t i o n predicates need not be all identical. For example, to distinguish between objects t h a t move and objects t h a t don't regardless of their m o t i o n type, we define t w o categories: " n o - m o t i o n " , consisting of the m o t i o n type fixed, and " m o t i o n " , consisting of all the other m o t i o n types. Behavior Parametrization and Periodicity Abstraction: These two operators are direct extensions of the local diagram operators. Regions are parameterized by t a k i n g i n t o account all m o t i o n labels, motion parameter relations, and intervals. Periodicity abstraction finds isomorphisms between subgraphs by using the extended definition of region similarity. T h e order in which the global operators are applied follows the same rationale of local operators: the global kinematic constraints simplifications are applied first d u r i n g the c o m p u t a t i o n of the global region diagram because they discard potential regions w i t h o u t computing t h e m explicitly. T h e n , Relevance-Set is applied to focus on the behaviors of relevant objects. Next, RegionDifference is applied to create a coarser behavioral de-

scription; if the number of regions does not decrease, these abstractions are unnecessary. Periodicity abstraction is applied at the end. As an example, consider the effects of global operators on the description the cylinder lock of Fig. 3. We assume that the key can only receive i n p u t motions along axis O, and that the only behaviors of interest are those of the key and the cylinder. A p p l y i n g the kinematic constraints simplifications to the local descriptions of the pairs key/lower pins, ( K , P i ) , discards all the regions in which K and Pi are not in contact, except for the region containing the initial placement. This eliminates the 3 5 — 1 potential global regions resulting from the pin position combinations when the key is outside the cylinder. This operator also eliminates all the regions where pins not in contact w i t h the key are in different positions as the key is inserted; there are 3x (3 4 -f 3 3 + 3 2 - f 3 1 ) = 360 such potential global regions. The Relevance-Set simplification merges all the regions specifying different pin positions but no difference in the angular position of the cylinder or the key. The resulting projected region diagram consists of three regions: O U T The key and the cylinder are not in contact. The key can both rotate and translate along O, and the cylinder cannot rotate. IN The key is inside the cylinder. The key translate along O, but cannot rotate. The cylinder cannot rotate either. U N L O C K The key is at the end of the cylinder. The key can rotate together w i t h the cylinder, but it. cannot translate. No further application of Region-Difference or Periodici t y abstraction is necessary.

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C o m p a r i n g T w o Mechanisms

Mechanism comparison is an example of a task that cannot be automated without simplifications and abstractions. It consists of determining when t w o mechanisms can be considered kinematically equivalent. The ability to compare two mechanisms is essential for evaluating design solutions and choosing existing mechanisms for redesign. To compare two mechanisms, we compare their behaviors, rather than their structure. For example, the two crank mechanisms in Fig. 5 are structurally very different; however, they both transform a continuous rotation into a reciprocating translation. But many of the details of this transformation are different: the number of parts, the motion ratios, etc. Thus, a direct comparison of the mechanisms' region diagrams will also indicate that they are different. The comparison algorithm uses the simplification and abstraction operators to match the descriptions. First, the region diagrams of both mechanisms are computed, incorporating the kinematic and dynamic constraints. T h e n , the Relevance-Set operator is applied to discard unnecessary behavioral information. The other operators are then applied in the order indicated in sections 3 and 4. After each application, the region diagrams

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m e n t e d a n u m b e r of local a b s t r a c t i o n s very similar to the ones presented in t h i s paper: L i n e a r i z a t i o n , Q u a l i t a t i v e , and Gap-closing a b s t r a c t i o n . H i s t r e a t m e n t of force a n d m o t i o n i s s i m i l a r t o our concept o f i n p u t m o t i o n . N o global abstractions are proposed. P r o d u c i n g q u a l i t a t i v e descriptions at m u l t i p l e levels of resolution is an i m p o r t a n t p r o b l e m . Recently, [Falkenhainer and F o r b u s , 88] described a scheme to b u i l d large-scale q u a l i t a t i v e models; [ M u r t h y , 88] proposed a scheme for d y n a m i c a l l y a d j u s t i n g t h e level of resolut i o n in w h i c h a device is analyzed. O u r research shares m a n y of t h e i r goals; however, their techniques are i n a d equate for k i n e m a t i c behaviors because t h e i r q u a l i t a t i v e states are defined as n - d i m e n s i o n a l r e c t a n g u l a r regions of q u a n t i t y spaces. T h i s d i s t i n c t i o n is t o o crude to acc o u n t for CS-boundaries t h a t p a r t i t i o n the space i n t o n o n - r e c t a n g u l a r regions. M o r e o v e r , t h e i r characterizat i o n does n o t account for different a p p r o x i m a t i o n s to the

(a)

(b)

exact CS-boundaries. Possible extensions and f u t u r e w o r k include: (1) i m p l e m e n t i n g the operators; (2) a p p l y i n g t h e operators t o d o m a i n s were the p r o b l e m can be f o r m u l a t e d w i t h parameter spaces, e.g., d y n a m i c s ; (3) e x p l o r i n g the role of the operators in design, a n d ; (4) classifying mechanisms for redesign.

References are c o m p a r e d , a n d a parameterized m a t c h between their regions i s a t t e m p t e d . W h e n the t w o diagrams m a t c h the a l g o r i t h m stops, p r o d u c i n g a c o m m o n d e s c r i p t i o n for b o t h region d i a g r a m s , and the list of operators necessary to achieve the equivalence. T h i s list, establishes the smallest set of c o n d i t i o n s necessary to d e t e r m i n e w h e n t w o mechanisms are equivalent (see [Joskowicz, 89b]). T h e t w o c r a n k mechanisms in F i g . 5 can be considered equivalent w h e n : (1) i n p u t m o t i o n s are o n l y applied to the d r i v i n g wheel; due to f r i c t i o n , the second assembly is not reversible, i.e., t h e t r a n s l a t i o n of the slider w i l l n o t cause the r o t a t i o n of the wheel ( k i n e m a t i c and d y n a m i c constraints) (2) the o n l y behaviors of interest are those of t h e slider and the d r i v i n g wheel (Relevance-Set); (3) the m o t i o n relations between the d r i v i n g wheel and the slider are q u a l i t a t i v e l y similar ( Q u a l i t a t i v e A b s t r a c t i o n ) (4) the gears backlash are negligible ( G a p - C l o s i n g A b s t r a c t i o n ) and (5) b o t h sliders' displacements are inside the desired o p e r a t i n g range ( I n t e r v a l P a r a m e t e r i z a t i o n ) .

5

Conclusion

F u t u r e I n t e l l i g e n t C A D systems m u s t b e able t o problem-solve using q u a l i t a t i v e descriptions at different levels of resolution. T h i s paper presented a set of operators to s i m p l i f y and abstract k i n e m a t i c descriptions derived f r o m c o n f i g u r a t i o n spaces. T o i l l u s t r a t e t h e i r use, we showed how m e c h a n i s m comparison can be done w i t h these operators. T h e need for a b s t r a c t i n g k i n e m a t i c descriptions derived f r o m c o n f i g u r a t i o n spaces was recognized independ e n t l y by Nielsen ( 1 9 8 8 b ) 4 . He developed and i m p l e 4

I am grateful to the anonymous reviewers for bringing this work to my attention.

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Knowledge Representation

[ F a l k e n h a i n e r a n d F o r b u s , 88] B. Falkenhainer and K. Forbus, "Setting up Large-Scale Qualitative Models'', Proc. of AAAI-88, St. Paul, Minnesota, 1988. [ F a l t i n g s , 87a] B. Fallings, "Qualitative Place Vocabularies for Mechanisms in Configuration Space", Phd. Diss, UIUCDCS-R-87-1360, U. of Illinois, 1987. [ F a l t i n g s , 87b] B. Faltings, "Qualitative Kinematics in Mechanisms" Proc. of 1JCAI-87, Milano, Italy, 1987. [ J o s k o w i c z , 87] L. Joskowicz, "Shape and Function in Mechanical Devices", Proc. AAAI-87, Seattle, 1987. [ J o s k o w i c z , 88] "Reasoning about Shape and Kinematic Function in Mechanical Devices" PhD. Diss, 402, Courant Institute, New York University, 1988. [ J o s k o w i c z , 89a] L. Joskowicz, "Reasoning about the Kinematics of Mechanical Devices", Int. Journal of Artificial Intelligence in Engineering, Vol 4, 1989. [ J o s k o w i c z , 89b] L. Joskowicz, "Mechanism Comparison and Classification for Redesign", I B M Technical Report, to appear, 1989. [ J o s k o w i c z a n d A d d a n k i , 88] L. Joskowicz and S. Ad danki, "From Kinematics to Shape: An Approach to Innovative Design", Proc. of AAAI-88, St. Paul, M i n nesota, 1988. [ L o z a n o - P e r e z , 83] 1 T. Lozano-Perez, "Spatial Planning: A Configuration Space Approach", IEEE Transactions on Computers, Vol. C - 3 2 , No. 2, 1983. [ M u r t h y , 88] S. M u r t h y , "Qualitative Reasoning at M u l t i ple Levels of Resolution", Proc. of AAAI-88, St. Paul, Minnesota, 1988. [ N i e l s e n , 88a] P. Nielsen, "A Qualitative Approach to Mechanical Constraint", Proc. of AAAI-88, St. Paul, M i n nesota, 1988. [ N i e l e s e n , 88b] P. Nielsen, "A Qualitative Approach to Rigid Body Mechanics", Phd. Diss, UIUCDCS-R-881469, U. of Illinois, November 1988.