Computer-Aided Mechanical Design Using Configuration Spaces

Transcription

1 Compter-Aided Mechanical Design Using Configration Spaces Leo Joskowicz Institte of Compter Science The Hebrew University Jersalem 91904, Israel Elisha Sacks (corresponding athor) Compter Science Department Prde University West Lafayette, IN 47907, USA Jne 24, 1999 Abstract The paper describes research in compter-aided design of mechanical systems sing configration spaces. The research addresses the core design task of rigid-body contact analysis and related tasks. Contact analysis is a comptational bottleneck in mechanical design, especially in systems with complex part shapes, tight fits, and contact changes. Manal analysis is error-prone and time-consming, whereas atomated analysis exceeds the capabilities of crrent design software. To address these problems, we have developed a general contact analysis method for planar mechanical systems based on configration space comptation. We have implemented a prototype design environment that integrates contact analysis with simlation, tolerance analysis, and visalization. The software helps designers stdy system fnction nder a range of operating conditions, find and correct design flaws, and optimize performance. Keywords: compter-aided design, mechanism theory, contact analysis, kinematics. To appear in IEEE Compters in Science and Engineering,

2 1 Introdction This paper describes or research in compter-aided design of mechanical systems sing configration spaces. Mechanical design is the task of devising an assembly of parts (a mechanical system) that performs a fnction reliably and economically. It is a biqitos activity with applications in mechanical, electrical, and biomedical engineering. Designers need to devise, analyze, and compare competing design prototypes to create good designs. Compter-aided design helps designers redce design time and improve qality by replacing physical prototypes with electronic ones. Or research addresses the core design task of rigid-body contact analysis and related tasks. We assme that the parts are rigid: they cannot change shape or overlap. This assmption is reasonable for most mechanical design tasks. Contact analysis determines the positions and orientations at which the parts of a system toch and the ways that the toching parts interact. The interactions consist of constraints on the part motions that prevent them from overlapping. The constraints are expressed as algebraic eqations that relate the part coordinates. For example, a rond ball on a flat floor obeys the constraint z r = 0 with z the height of its center point (a position coordinate) and r its radis. The constraints are a fnction of the shapes of the toching part featres (vertices, edges, and faces), hence they change when one pair of featres breaks contact and another makes contact. Contact analysis is a core design task becase contacts are the physical primitives that make mechanical systems ot of collections of parts. Systems perform fnctions by transforming motions via part contacts. The shapes of the interacting parts impose constraints on their motions that largely determine the system fnction. Designers perform contact analysis to derive this evolving seqence of contacts and motion constraints. The reslts help them simlate the system fnction, find and correct design flaws, measre performance, and compare design alternatives. We illstrate contact analysis and its role in design on the ratchet mechanism shown in Figre 1. The mechanism has for moving parts and a fixed frame. The driver, link, and ratchet are attached to the frame by revolte joints. (A revolte joint is a cylindrical pin on one part that fits in a matching cylindrical hole in the other part, ths restricting the part motions to rotation abot the cylinder axis.) The pawl is attached to the link by a revolte joint and is attached to a spring (not shown) that applies a conterclockwise torqe arond the joint. A motor rotates the driver with constant anglar velocity, casing the link pin to move left and right. This cases the link to oscillate arond its rotation point, which moves the pawl left and right. The leftward motion pshes a ratchet tooth, which rotates the ratchet conterclockwise. The rightward motion frees the pawl tip from the tooth, which allows the spring to rotate the pawl to engage the next tooth. Contact analysis validates the intended fnction by determining if the link oscillates far enogh, if the pawl pshes the ratchet teeth far enogh, if the system can jam, and so on. Revolte joints are standard and easy to analyze, althogh the combined effect of several joints is complex. The driver/link pair is harder to analyze becase the link pin interacts with the inner and oter driver profiles. The ratchet/pawl pair is mch harder yet becase the part shapes are complex and becase the pawl can translate horizontally, translate vertically, and rotate, whereas the other parts jst 2

3 pawl ratchet link driver (a) (b) (c) Figre 1: Ratchet mechanism: (a) pawl advancing ratchet; (b) pawl flly advanced; (c) pawl retracting. White circles indicate revolte joints. rotate. Contact analysis of the overall system is hardest of all becase we mst validate the intended interactions among all the parts, sch as the indirect relation between the driver and the ratchet, and mst rle ot interference, sch as the pawl hitting the frame. Contact analysis is a comptational bottleneck in mechanical systems with many potential part contacts. The complexity grows rapidly as the nmber of parts increases. A pair of toching parts interacts via contacts between featre pairs. Hndreds of featres per part is the norm, which leads to thosands of potential contacts per pair and to a combinatorial growth in system contacts. Or research shows that this complexity is common in modern mechanisms. Contact changes occr by design in 65% of the 2500 mechanisms in an engineering encyclopedia [1]. Examples inclde gears, cams, ratchets and cltches. Manfactring variation often introdces nintended, complex contacts into a system whose nominal fnction has simple contacts. For example variation in revolte joints cases play: the pin can translate as well as rotate becase it is smaller than the hole. Designers need to analyze these variations to ensre correct fnction. This process is called tolerance analysis. Manal contact analysis of complex systems is error-prone and time-consming, whereas atomated analysis exceeds the capabilities of crrent design software. This software consists mainly of simlators for systems whose parts interact via standard joints, sch as linkage mechanisms and robot maniplators [2]. It does not directly handle systems with non-standard joints, contact changes, and tight fits. To address these problems, we have developed a general contact analysis method for planar mechanical systems, which accont for over 90% of mechanisms. We have implemented a prototype design environment that integrates contact analysis with simlation, tolerance analysis, and visalization. The software helps designers stdy system fnction nder a range of operating conditions, find and correct design flaws, and optimize performance. The contact analysis method is based on configration space comptation. Configration space is a geometric representation of rigid body interaction that has seen extensive comptational se in robot motion planning [3]. We have fond that it is an effective tool for contact analysis. The configration space of a mechanical system describes all possible part interactions. It encodes 3

4 y a a θa a ψ v y b b θ b b x a x (a) b (b) Figre 2: Pairwise configrations: (a) absolte coordinates and (b) relative coordinates. qantitative information, sch as part motion paths, and qalitative information, sch as system failre modes. It provides a framework within which diverse design tasks can be performed. We have developed a configration space comptation algorithm for systems with crved parts, generalized it to toleranced parts, and applied it to mechanical design. This paper is a srvey of or research on contact analysis. We describe the configration space representation, explain its vale as a contact analysis tool, and illstrate it on simple examples. In so doing, we present the big pictre behind the diverse algorithms that appear in prior pblications. We assess the strengths and weaknesses of or approach for practical design tasks and identify isses for ftre research. 2 Configration space We perform contact analysis on a mechanical system by compting a configration space for each pair of parts. The prpose of the analysis is to identify the pairs of part featres that toch in some system configration, to compte the motion constraints for every pair, and to compte the configrations in which contacts change. This section explains what configration space is and how it spports contact analysis. We attach reference frames to the parts and define the configration of a part to be the position and orientation of its reference frame with respect to a fixed global frame. Figre 2a shows parts a and b, their reference frames, and their configrations (x a ; y a ; a ) and (x b ; y b ; b ). The configration space of the pair is the Cartesian prodct, (x a ; y a ; a ; x b ; y b ; b ), of the part configrations. The configration space coordinates represent the six independent motions of the parts, called degrees of freedom. As the parts move, the configration traces a path in configration space. Configration space partitions into three disjoint sets that characterize part interaction: blocked space where the parts overlap, free space where they do not toch, and contact space where they toch. The free and blocked spaces are open sets whose common bondary is contact space. This implies that the first two have the same dimension as the configration space, whereas the 4

5 dimension of the third is one lower. Intitively, free and blocked space are open becase disjoint or overlapping parts remain so nder all small motions, whereas contact space is closed becase toching parts separate or overlap nder some small motions. We illstrate these concepts with the simple example of a block that moves in a fixed frame. In Figre 3(a), the frame is fixed at the global origin with orientation 0, so we can ignore its coordinates from the configration space and consider only the block coordinates (; v; ). We first assme that the block translates in the displayed orientation withot rotating, which yields a two-dimensional configration space. The gray region is blocked space, the white region is free space, and the black lines are contact space. The dot in free space marks the displayed position of the block. Free space divides into a central rectangle where the block is inside the frame, an oter region where it is otside, and a narrow connecting rectangle where it is partly inside. The contact crves (lines in this case) bonding these regions represent contacts between the vertices and edges of the block and the frame. Changing the orientation of the block yields configration spaces with different topologies (Figre 3b, c). The free space consists of disconnected inside and otside regions becase the block does not fit throgh the frame opening. We now consider the same example, bt with the block orientation varying from to radians, The configration space becomes three dimensional with rotation coordinate (Figre 4). One way to visalize this space is as a stack of planar slices along the rotation axis. Each slice is the configration space of a block that translates at a fixed orientation, sch as the three examples above. The fll space is the nion of the slices. The free space consists of an oter region, two inner regions, and two connecting channels near = =2 where the block is nearly vertical. The oter tbe is the nion of the oter regions, the inner tbe is the nion of the inner rectangles, and the channels are the nion of the connecting regions. Blocked space is the region between the tbes and otside the channels. We can model general planar pairs with three-dimensional configration spaces even thogh they have six degrees of freedom. The reason is that the part contacts are invariant nder rigid motions of the pair. In other words, the relative configration of the parts determines the contacts. We compte the configration space of part a with respect to the reference frame of part b, which is eqivalent to fixing b in the (0; 0; 0) configration. The relation between the absolte and relative coordinate systems is given by = (x a x b ) cos b + (y a y b ) sin b v = (y a y b ) cos b (x a x b ) sin b = a b : (1) Figre 2 contrasts the two coordinate systems. In the example, the block is a, the frame is b, and (; v; ) = (x a ; y a ; a ) becase x b ; y b ; b = 0. Whatever its dimension, the configration space of a pair is a complete representation of the part contacts, so any contact qestion is answerable by a configration space qery. Testing if parts overlap, do not toch, or are in contact in a given configration corresponds to testing if the configration point is in blocked, free, or contact space. Contacts between pairs of featres 5

6 contact space block free space v frame blocked space (a) v v (b) v v (c) Figre 3: A translating block moving arond a fixed frame. 6

7 v ψ Figre 4: Contact space for block with three degrees of freedom shaded with a niqe color for each contact patch. 7

8 correspond to contact patches (crve segments in two dimensions and srface patches in three). The patch geometry encodes the motion constraint and the patch bondary encodes the contact change conditions. Part motions correspond to paths in configration space. A path is legal if it lies in free and contact space, bt illegal if it intersects blocked space. Contacts occr at configrations where the path crosses from free to contact space, break where it crosses from contact to free space, and change where it crosses between neighboring contact patches. The configration space representation generalizes from pairs of parts to systems with more than two parts. A system of n planar parts has a 3n-dimensional configration space whose points specify the n part configrations. A system configration is free when no parts toch, is blocked when two parts overlap, and is in contact when two parts toch and no parts overlap. System configration spaces allow s to analyze mlti-part interactions, sch as the motion relation between the driver and the ratchet in the ratchet mechanism. They are often impractical to compte and are not needed for the design tasks that we discss. 3 Configration space comptation Configration space comptation has algebraic and combinatorial components. The algebraic task is to derive the contact constraint for a pair of featres and to compte the reslting contact patch. The combinatorial task is to compte the configration space partition, which is determined by contact patch intersections. Both tasks are comptationally expensive and difficlt to implement for general shapes. The robotics literatre contains many specific configration space comptation algorithms (see [3] for the basic algorithms). That research provides practical algorithms for pairs of polygons and contact constraints for polyhedra. These algorithms do not extend to crved parts becase they rely on the special strctre of polygonal contact spaces, which are made p of rled srface patches generated by vertex/edge contacts. Polygons are fine for path planning, which is the primary robotics application of configration space, bt are inappropriate for mechanical design where precise motion constraints between crved featres are crcial to system fnction. We have developed a fast, robst configration space comptation algorithm for pairs of planar parts whose bondaries are comprised of line segments and of circlar arcs. These featres sffice for most engineering applications. The program distingishes between standard joints, fixed-axes pairs, and general pairs. The standard joint types are revolte, prismatic, and sliding. We have seen revolte joints in the ratchet mechanism; the other types are described in engineering texts. Each standard joint imposes a permanent contact that indces a fixed set of motion constraints, which the program retrieves from a table. Fixed-axes pairs consists of two parts with one degree of freedom apiece. They are analyzed by a fast algorithm that exploits their special strctre. Standard joints and fixed-axes pairs accont for over 90% of planar pairs based on or srvey of 2500 mechanisms [1]. The remaining pairs are analyzed by a general algorithm. Fixed-axes pairs have two-dimensional configration spaces whose coordinates are the motion 8

9 ω 0.15 π θ π Figre 5: Driver/link pair and its configration space. coordinates of the two parts. For example, Figre 5 shows the configration space of the driver/link pair. The configration space coordinates are the driver orientation and the link orientation!. The pper and lower contact crves represent contacts between the cylindrical pin and the oter and inner cam profiles. The free space is the region in between. As the driver rotates from = to = 0, its inner profile pshes the link pin right, which rotates the link conter-clockwise from! = 0:47 radians to! = 0:105 radians. As the driver rotates from = 0 to =, the pin breaks contact with the inner profile and makes contact with the oter one, which plls it left and rotates the link clockwise. The configration follows the lower contact crve from = to 0, travels horizontally throgh free space ntil it hits the pper contact crve, and follows it to. The fixed-axes program [4] comptes these two-dimensional configration spaces. The contact crves are obtained from a hand-compted table with one entry for each combination of featre types (line segments, arcs, and points) and motion types (horizontal translation, vertical translation, and rotation), for example a rotating line segment and a circle translating horizontally. The program enmerates the featre pairs, generates their contact crves from the table, discretizes them to an inpt tolerance (0:001 relative accracy in typical engineering applications, and comptes the configration space partition that they indce. The connected component that contains the initial configration is the reachable portion of the configration space. The program handles any realistic pair in well nder one second (100,000 contacts in 0.1 seconds on a workstation). The general program [5] comptes three-dimensional configration spaces. The contact patches are implicit, the patch bondary crves are parametric, and the contact space is in bondary representation. The patches and crves are obtained from a hand-compted table as before. The partition is compted with a planar line sweep by dimension redction. The entire comptation is exact. The rnning time is nder one minte for every pair that we have tested, inclding ones with 10,000 contacts. Figre 6 shows the three-dimensional configration space of the ratchet moving relative to the pawl. Althogh hard to visalize, it encodes a fll contact analysis. Let s examine the slice in 9

10 v 30 ψ (a) (b) (c) Figre 6: (a) Ratchet/pawl pair; (b) configration space slice at = 0:277 radians; (c) contact space; the inner region is blocked space, the oter region free space. -30 v the figre, which shows how the ratchet translates in the displayed orientation. The dot marks the displayed position of the ratchet relative to the pawl. It lies on a contact crve that represents contact between the pawl tip and the side of a ratchet tooth. The right end of the crve is the intersection point with a second contact crve that represents contact between the left corner of the pawl and the next tooth conter-clockwise. The ratchet can maintain this contact while translating right ntil the second contact occrs and frther translation is blocked. Rotation can only be expressed in the fll configration space. 4 Dynamical simlation Configration spaces spport a novel form of dynamical simlation that is well-sited to mechanical design. Dynamical simlation consists of compting the motions of the parts in accordance with Newton s laws. It is an important design tool for those aspects of system fnction that depend on dynamical effects, sch as inertia, friction, and gravity. It provides loads for finite-element analysis: a nmerical analysis of part interactions that lie otside the rigid-body idealization, sch as deformation and stress. We illstrate simlation on the ratchet mechanism. The external forces are a motor that rotates the driver, a torsional spring that rotates the pawl conterclockwise, and the weight of the pawl. A short simlation shows that the first cycle appears correct (Figre 1). Bt a longer simlation reveals a problem: the ratchet rotates faster at every cycle becase of the repeated pawl impacts. We can correct this problem by adding an external load or internal damping to the ratchet. Frther simlation tests how strong a spring is needed to maintain the pawl/ratchet contact with varios driving torqes. Contact analysis is a prereqisite for simlation becase contacts create forces that effect part motion. The simlator needs to know which part featres toch at every instant and when contact 10

11 changes occr. Given this information, it can compte the contact forces, combine them with the external forces, compte the part accelerations from Newton s laws, and nmerically integrate them to obtain the part configrations and velocities at the next time step. At each time step, the simlator checks for contact changes since the previos step, which force it to back p to the change time and pdate the contact eqations. Manal analysis is practical for systems with permanent contacts [6] and is implemented in commercial simlators [2], bt atomated analysis is crcial for systems with many contact changes. We have developed a simlator that ses configration spaces for atomated contact analysis [7]. The configration spaces of the interacting pairs are compted before the simlation. At each time step, the simlator qeries them for the contact data for contact force comptation. It tests for part collisions and contact changes between steps by qerying the configration spaces for transitions between free and contact space or between contact patches. We have simlated systems with tens of moving parts at interactive speeds. The largest simlation to date is a chain assembly with two gears, a 34-link chain connected by pin joints, and 68 link/gear higher pairs. An alternate approach, developed primarily in graphics research [8], is to test all pairs of parts for collisions at each time step. The test can be extremely fast even for large systems. The disadvantages for mechanical systems are that crrent algorithms do not handle crved parts and are inefficient when the parts are close together and interact often. The main advantage of collision detection algorithms is that they handle three-dimensional parts. 5 Tolerance analysis Configration spaces spport atomated tolerance analysis of mechanical systems. The task is to compte the variation in the system fnction de to manfactring variation in the parts. If we se parametric part models, the part variations can be represented as intervals arond the nominal parameter vales. In the ratchet mechanism, the parameters wold inclde the driver radis and eccentricity, the link length, the pawl length and tip angle, and the ratchet tooth length and slant. The analysis can be qalitative, qantitative, or statistical. Qalitative analysis tells s if the part parameter variations can case nintended contact effects, for example if the pawl can hit the frame or can fail to advance the ratchet. This analysis mst be performed first becase it provides the contact constraints for the other types. Qantitative analysis gives s the derivatives of the part configrations with respect to the parameters, which allows s to compte the maximal error in system fnction. Statistical analysis estimates the fraction of mechanisms that will fail. Tolerance analysis prespposes contact analysis becase the variation in the system fnction arises from variations in the part contact constraints. We need to know which contacts occr at each stage of the work cycle and how their constraints depend on the part parameters. Manal analysis is often infeasible becase of the many contacts and the complex relations between part parameters and contact constraints. We have developed a tolerance analysis algorithm for planar systems based on a generalization of configration space to parametric parts [9]. The algorithm 11

12 ω 0.12 θ 0.75 Figre 7: Detail of driver/link generalized configration space. performs qantitative and statistical analyses and helps designers perform qalitative analysis. It analyzes systems with 50 to 100 parameters in a few mintes, which permits interactive tolerancing of detailed fnctional models. Figre 7 shows a generalized configration space for the driver/link pair of the ratchet mechanism: colored crves sperimposed on the nominal configration space. The red and green crves are pper and lower bonds on the variation in the contact space de to the part variations. If the part parameters are in their tolerance intervals, the contact space mst be between these crves. The channel between the top green crve and the bottom red crve represents the worst-case pin clearance. It is smallest at = 0 where the pin is at its rightmost position and largest where the pin is at its leftmost position. Increasing the part tolerances brings these crves closer. When they meet, the the qalitative mechanism fnction alters (a failre mode) becase the pin cannot complete its cycle. 6 Conclsion We have seen that configration space comptation is a practical algorithm for contact analysis of planar mechanical systems and for the related design tasks of dynamical simlation and tolerance analysis. We have tested the algorithms on dozens of mechanical systems, inclding the gearshift of an atomotive transmission and varios micro-mechanisms. Or ltimate goal is to develop an atomated configration space-based program to spport the design engineer in all the steps of design. We are crrently psing research in spatial contact analysis, tolerance synthesis, interactive 12

13 parametric design, and configration space visalization. The next step in or contact analysis research is configration space comptation for pairs of three-dimensional parts. We have already developed an algorithm for pairs of rigid parts that move along fixed spatial axes and whose shapes are formed by planes, cylinders, or spheres [10]. Since we do not believe that the general spatial case, which reqires compting six-dimensional configration spaces (each part has three rotations and three translations) is tractable, we are focsing on on specialized algorithms for important special cases. For tolerance synthesis, we plan to se or tolerance space approach to compte the variability cost of each contact in the nominal contact seqence and atomatically derive the eqations and optimization fnctions for nonlinear optimization to obtain the best tolerance allocation. We will se the parametric contact fnctions and sensitivity analysis derived in or existing tolerance analysis algorithm. For parametric design we plan to develop algorithms that allow interactively inverting the mapping from parameter vales to configration spaces. The goal is to allow the designer to modify a configration space interactively while the design program pdates the assembly to realize the changing kinematics. Compter visalization is a powerfl tool for nderstanding the qalitative strctre of complex configration spaces. We are developing cstom visalization software that elcidates this strctre with graphics techniqes, sch as shading and color, and by correlating the configration space geometry with the system kinematics, for example by highlighting the part featres that generate a selected contact patch [11]. Acknowledgments Sacks is spported by NSF grants CCR and CCR and by the Prde Center for Comptational Image Analysis and Scientific Visalization. Joskowicz is spported by a grant from the Athority for Research and Development, The Hebrew University of Jersalem, Israel. Sacks and Joskowicz are spported by a Ford University Research Grant, the Ford ADAPT200 project, and by grant 98/536 from the Israeli Academy of Science. References [1] Leo Joskowicz and Elisha Sacks. Comptational kinematics. Artificial Intelligence, 51: , [2] W. Schiehlen. Mltibody systems handbook. Springer-Verlag, [3] Jean-Clade Latombe. Robot Motion Planning. Klwer Academic Pblishers,

14 [4] Elisha Sacks and Leo Joskowicz. Comptational kinematic analysis of higher pairs with mltiple contacts. Jornal of Mechanical Design, 117: , [5] Elisha Sacks. Practical sliced configration spaces for crved planar pairs. International Jornal of Robotics Research, 17(11), [6] Edward J. Hag. Compter-Aided Kinematics and Dynamics of Mechanical Systems, volme I: Basic Methods. Simon and Schster, [7] Elisha Sacks and Leo Joskowicz. Dynamical simlation of planar systems with changing contacts sing configration spaces. Jornal of Mechanical Design, 120: , [8] Ming C. Lin, Dinesh Manocha, Jon Cohen, and Stefan Gottschalk. Collision detection: Algorithms and applications. In Jean-Pal Lamond and Mark Overmars, editors, Algorithms for Robotic Motion and Maniplation. A. K. Peters, Boston, MA, [9] Elisha Sacks and Leo Joskowicz. Parametric kinematic tolerance analysis of general planar systems. Compter-Aided Design, 30(9): , [10] Iddo Drori, Leo Joskowicz, and Elisha Sacks. Spatial contact analysis of fixed-axes pairs sing configration spaces. In IEEE Conference on Robotics and Atomation, [11] Elisha Sacks and Leo Joskowicz. Configration space visalization for mechanical design. In Proceedings of Visalization 98 Conference, Research Triangle Park, NC,