Coping with Friction for Non-penetrating Rigid Body Simulation

Transcription

1 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 Coping with Friction for Non-penetrating Rigid ody Siulation Daid araff Progra of Coputer Graphics Cornell Uniersity Ithaca, NY bstract lgoriths and coputational coplexity easures for siulating the otion of contacting bodies with friction are presented. The bodies are restricted to be perfectly rigid bodies that contact at finitely any points. Contact forces between bodies ust satisfy the Coulob odel of friction. traditional principle of echanics is that contact forces are ipulsie if and only if non-ipulsie contact forces are insufficient to aintain the non-penetration constraints between bodies. When friction is allowed, it is known that ipulsie contact forces can be necessary een in the absence of collisions between bodies. This paper shows that coputing contact forces according to this traditional principle is likely to require exponential tie. n analysis of this result reeals that the principle for when ipulses can occur is too restrictie, and a natural reforulation of the principle is proposed. Using the reforulated principle, an algorith with expected polynoial tie behaior for coputing contact forces is presented. Categories and Subject Descriptors: I.3.5 [Coputer Graphics]: Coputational Geoetry and Object Modeling; I.3.7 [Coputer Graphics]: Three-Diensional Graphics and Realis dditional Key Words and Phrases: dynaics, friction, siulation, NP-coplete 1. Introduction The synthesis of realistic otion is one of the goals of coputer graphics. Recently, uch attention has been gien to physically based siulation ethods, and in particular, rigid body siulation. To achiee realis, siulations ust incorporate the effects of friction between contacting bodies. If the total nuber of contact points is sall, for instance one to four, the effects of friction are easily coputed. Howeer, as the nuber of contact points grows, the proble becoes considerably ore challenging. Siulation algoriths with exponential (in the nuber of contact points) running ties are known[5] but are ipractical for probles inoling as few as 10 to 15 contact points. In order to ake rigid body siulations with friction practical for coputer graphics, efficient, polynoial tie algoriths are needed. This paper considers the probles of coputing friction forces for configurations of perfectly rigid bodies with a finite nuber of contact points. For polyhedral bodies, only the ertices of the line segent and polygonal contact regions are considered as contact points. Unless otherwise stated, it is assued that This is an electronic reprint. Perission is granted to copy part or all of this paper for noncoercial use proided that the title and this copyright notice appear. This electronic reprint is 1994 by CMU. The original printed paper is 1991 by the CM. uthor address (May 1994): Daid araff, School of Coputer Science, Carnegie Mellon Uniersity, Pittburgh, P 15213, US. Eail: baraff@cs.cu.edu bodies are not colliding at any contact point. No restriction is placed on the allowable sliding otion between bodies at contact points. Forces at contact points are classified as either noral or friction forces. Noral forces preent inter-penetration by acting perpendicularly to the contact surfaces. Friction forces act tangentially to the contact surfaces and oppose slipping otion. The friction force at a contact point is called dynaic friction if the two bodies are slipping at the contact point; otherwise, the friction force is called static friction. The contact forces (the noral and friction forces) ust satisfy the Coulob odel of friction. The Coulob odel of friction is a well accepted epirical relationship between the noral and friction force at a contact point. n iportant first step to coping with the probles of friction is understanding the siulation behaior specified by the Coulob odel of friction. We need to know both what kind of result the odel specifies and the degree of difficulty in coputing that result. When coputing contact forces, a principle of rational echanics called the principle of constraints [9] is usually accepted. The principle of constraints states that constraints should be satisfied by non-ipulsie forces if possible; otherwise, ipulsie forces should be used to satisfy constraints. (Ipulsie forces, or ipulses, hae the units of ass ties elocity and discontinuously change elocities; ipulses ost coonly arise when bodies collide. Non-ipulsie forces, or just forces, hae the units of ass ties acceleration and cannot produce elocity discontinuities.) The first result of this paper is a proof that coputing friction forces according to the principle of constraints is likely to require exponential tie (section 5). Under the Coulob friction odel, een in the absence of collisions it is soeties necessary to introduce ipulses between contacting bodies to preent inter-penetration. dopting the principle of constraints requires that a particular behaior, non-ipulsie contact forces, be searched for aong possibly exponentially any other choices, wheneer possible. In foral ters, we will proe that deciding if non-ipulsie contact forces are sufficient to preent interpenetration is NP-coplete. Essentially, this eans that an efficient (that is, polynoial tie) algorith for coputing contact forces is widely belieed not to exist. (See Garey and Johnson[7] for a discussion on P, NP, NP-coplete and NP-hard probles). Howeer, the preference for non-ipulsie behaior is neither necessary nor justified. Using insights fro the NPcopleteness results of section 5, section 6 presents a physical odel for contact that argues against the principle of constraints. We will use this odel to reforulate the proble of coputing contact forces. Using the reforulated proble, we present an efficient algorithic siulation ethod for dealing with dynaic friction. The algorith has an expected running tie that is polynoial in the nuber of contact points of the configuration. This is the first efficient algorith we know of for coputing dynaic 31

2 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 friction forces. s a first step towards dealing with both static and dynaic friction, we present two preliinary approaches for coputing static and dynaic friction forces (section 8). The first approach approxiates both static and dynaic friction by using the general algorith for dynaic friction. The second approach uses an iteratie technique to copute static and dynaic friction forces; howeer, conergence is not guaranteed. 2. Definitions For configurations without friction, a alid set of contact forces is a set of noral forces satisfying three conditions. First, the noral force at each contact point ust be oriented to "push" the bodies apart. Second, the noral forces ust be sufficient to preent inter-penetration between bodies. Third, if two bodies are separating at a contact point, the noral force at the contact point ust be zero. For configurations without friction, a alid set of contact forces exists for any configuration of bodies. lthough a alid set of contact forces is not necessarily unique for frictionless configurations, all alid contact forces yield the sae accelerations of the bodies in the configuration[4]. Contact forces for frictionless configurations with n contact points can be found by forulating and soling a conex quadratic progra (QP) of n ariables. Methods for forulating this QP for bodies coposed of polyhedra and cured surfaces hae been presented in [1, 2, 6, 12]. Conex QP s with n ariables can be soled in tie polynoial to n and in practice are soled by algoriths whose worst case behaior is exponential but whose expected running tie is polynoial[14]. Configurations with friction are ore coplicated. Contact forces with friction are alid if they satisfy both the preious three conditions for noral forces and the Coulob friction odel (sections 4 and 8). Valid contact forces for configurations with just dynaic friction (and no static friction) can be found, as in the frictionless case, by coputing the solution to a QP. Unlike the frictionless case though, the QP associated with a configuration inoling dynaic friction is not necessarily conex. The existence of a practical solution ethod for non-conex QP s is considered unlikely, because soling non-conex QP s is NP-hard. dditionally, it is possible that the QP for a configuration with dynaic friction ay not een hae a solution. lthough the Coulob friction odel is well accepted, it has been known for at least a century that configurations of rigid bodies with dynaic friction exist that hae no alid set of contact forces. We call such a configuration inconsistent. Conersely, there are also configurations with dynaic friction where neither the set of alid contact forces nor the accelerations resulting fro those contact forces are unique. Such a configuration is called indeterinate. (See sections 4.1 and 4.2). 3. Preious Work Wang and Mason[16] present a detailed discussion on single contact point collisions inoling friction; in particular, ethods for coputing the contact ipulse resulting fro the collision are described. (We will not consider the general proble of collisions inoling friction in this paper). Mason and Wang[13] discuss inconsistent configurations and explain how to resole the inconsistency by applying ipulsie contact forces to the configuration. Howeer, it is first necessary to identify configurations as inconsistent. s we will show, this turns out to be a difficult proble. paper by Lötstedt[12] discusses a siulation ethod that aoids inconsistency by odification of the friction law. Lötstedt s ethod changes the Coulob odel into a relation between noral forces fro the preious tie step and friction forces fro the current tie step. Lötstedt s ethod approxiates both dynaic and static friction by soling a conex QP. It is not clear that Lötstedt s ethod can always be initialized so that it is nuerically stable. It is also unclear how to perfor such an initialization efficiently. 4. Contact Force Model We begin by considering configurations with only dynaic friction. Static friction is not considered until section 8. This section introduces a special-case of a single contact point configuration (figure 1). This configuration, and inor ariations of it, will be used seeral ties throughout this paper. In figure 1, body is a thin rod of length two with a syetric ass distribution that contacts body at a single contact point. ody (the "base") is fixed. Variables p a contact point ṗ a contact point elocity nˆ unit surface noral tˆ unit surface tangent s ass I s oent of inertia s angular elocity µ coefficient of friction g graitational acceleration Relations I = 16 θ=72 16(cos 2 θ µcosθsinθ) = 2 nˆ tˆ ṗ a p a θ (graity) Figure 1. one contact point configuration with dynaic friction between a thin rod and a fixed base. y choosing s angular elocity and the agnitude g of the graity force acting on, an indeterinate and an inconsistent configuration can be produced. This particular exaple can be found in a nuber of papers; for exaple, Lötstedt[11], Erdann[5], or Mason and Wang[13]. For a gien alue of, the linear elocity of is chosen such that the point p a on has a non-zero elocity tangent to, and zero elocity noral to. The unit ector nˆ is noral to the surface of. The unit ector tˆ is tangent to the surface of, and is directed opposite to the otion of the point p a ; nˆ and tˆ are perpendicular. The particular alues of I, θ and µ (µ 3 4) gien in figure 1 are soewhat arbitrary; these alues are chosen to siplify later coputations. The Coulob odel of friction states that since p a is sliding across, a friction force in the direction tˆ acts on. (n equal and opposite friction force acts on, but is fixed.) If the noral force acting on has agnitude f, then the Coulob friction odel states that the friction force has a agnitude of µf (figure 2). The net contact force acting on is fnˆ +µftˆ = f(nˆ +µtˆ). (1) 32

3 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 (a) f = 0 (b) f = nˆ fnˆ fnˆ +µftˆ tˆ µftˆ Figure 2. Noral and friction forces acting on. What effect do the contact and external force hae on? In appendix, the coponent of acceleration of the point p a, noral to, is found to be nˆ.. ṗ a = f + ( 2 sinθ g). (2) This configuration has the odd property that as the noral force agnitude f is increased, the point p a is accelerated ore strongly towards! Geoetrically, the direction of the net contact force f(nˆ +µtˆ) does not change as f is increased. Howeer, as f is increased, the torque due to friction causes p a to angularly accelerate downward. The noral force fnˆ also causes the center of ass of, and thus p a, to accelerate upwards, but not fast enough to oercoe the downwards acceleration due to the torque. The net result is that increasing f decreases the alue of.a. (See appendix for details). 4.1 n Indeterinate Configuration To produce an indeterinate configuration, let and g satisfy 2 sinθ g=1. Then equation (2) becoes nˆ.. ṗ a = f + 1. (3) Recall that alid contact forces satisfy three conditions. The first condition, that the noral force "push" bodies apart is siply f 0. The second condition, that contact forces preent interpenetration, requires the acceleration of p a in the nˆ direction to be non-negatie. This yields the constraint.a 0. The last condition is that if the bodies are separating, the noral force ust be zero. Since the bodies are separating if and only if.a is strictly positie, this condition ay be written as f.a = 0. For 2 sinθ g=1 and using equation (2), the aboe three conditions are f 0, f and f( f + 1) = 0. (4) The alid contact forces are gien by the solution of equation (4); f = 0 and f =. For the f = 0 solution,.a = 1. In this solution, the centripetal acceleration of ṗ. a is stronger than the force of graity pulling down; thus, erely continues its rotation and the point p a oes off of (figure 3a). In the second solution, f = and.a = 0. noral force of nˆ and a friction force of µtˆ act on. The torque generated by friction balances the centripetal acceleration of p a ; as a result, and do not break contact (figure 3b). Note that the only alid alues of f are f = 0orf=. Since the solutions produce different accelerations for, the configuration is indeterinate. p a µtˆ Figure 3. (a) The contact force between and is zero. p a rotates to the left and up, breaking contact with. (b) The noral and friction forces balance graity and centripetal acceleration; p a oes horizontally and aintains contact with. 4.2 n Inconsistent Configuration Now suppose that =0 and s linear elocity is opposite tˆ (figure 4). Then the condition.a 0is. nˆ. ṗ a = f g 0. (5) Howeer, if g> 0 (figure 4a), then no positie alue of f can preent p a fro accelerating downwards and thus interpenetrating; that is, equation (5) cannot be satisfied by any f>0. This eans that the configuration is inconsistent. The existence of such a configuration ay see counter-intuitie; howeer, we will hae ore to say on this phenoenon in section 6.1. Note that the alue of g is crucial. If g = 0, so that no external force acts on, then f = 0 becoes the (unique) alid contact force (figure 4b). ny positie alue of f for this configuration causes inter-penetration. Figure 4b corresponds to p a "skiing" horizontally oer, with neither a noral force nor a friction force exerted on. Ifg becoes een slightly positie howeer, the configuration is inconsistent. Note that the requireent that be fixed is not crucial. If is assie copared to, then inconsistency occurs if an external force acts on to accelerate it towards, or ice ersa. (a) no solution for f µftˆ fnˆ (b) f = 0 Figure 4. (a) n inconsistent configuration. For any f 0, p a is accelerated downwards into. (b) The configuration has a unique solution of f = 0 when graity is reoed; skis along the surface of. nˆ p a 33

4 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July n NP-coplete Class of Configurations We define the frictional consistency proble as the proble of deciding if a gien configuration is consistent. In this section, we proe that the frictional consistency proble is NPcoplete. We begin by showing that the frictional consistency proble lies in NP and then show that the frictional consistency proble is NP-hard. lthough the configurations constructed in this section see contried (and arguably are), the NP-hardness result has grae iplications een when inconsistency is not a concern during siulation. Definition. n instance of the frictional consistency proble is a configuration C of bodies that contact at n distinct contact points. The physical properties of each body (ass, oent of inertia, linear and angular elocity, position and orientation, and external forces) are described by rational nubers. The specifics of a contact point (position, coefficient of friction, surface noral) are also described by rational nubers. The relatie otion between bodies at contact points with friction is non-zero in the direction tangent to the contact surface and zero in the direction noral to the contact surface. The notation C = k eans that configuration C is describable in k bits. Clearly k>n. Theore 1. The frictional consistency proble lies in NP. Proof. Gien an instance of C, a QP of size n with the following two properties exists. (1) If C is consistent, then an n-ector x that is a solution to the QP exists. The set of contact forces such that the agnitude of the noral force at the ith contact point is x i is a alid set of contact forces for the configuration C. (2) Otherwise, if C is inconsistent, the QP has no solution. The specifics of constructing the QP can be found in [6]. The nuerical quantities in the QP are coputed fro the rational entries of C in a total of O(n 3 ) arithetical operations. The QP can therefore be constructed in tie polynoial to k. Vaasis[15] has recently shown that quadratic prograing lies in NP. It follows fro this that deciding frictional consistency is also in NP. In order to show that deciding frictional consistency is NP-hard, we reduce the NP-coplete proble "subset su" to the frictional consistency proble. Definition. n instance of the subset su proble is a pair (,S) where = {a 1,...,an } is a set of positie integers and S is a single positie integer. subset su instance (,S) is satisfiable if there exists a subset such that Σ a a = S. (6) Deciding if an instance of the subset su proble is satisfiable is an NP-coplete proble[7]. To show that deciding frictional consistency is NP-hard we take an arbitrary instance (,S) of the subset su proble and construct (in polynoial tie) a configuration of bodies C. The configuration C will hae the property that C is consistent if and only if (,S) is satisfiable. Theore 2. Deciding frictional consistency is NP-hard. Proof. Consider the configuration of figure 5. ody of figure 5 is initially at rest and is positioned by four fixed triangular wedges that contact without friction. ody is therefore free to oe horizontally, but can neither rotate nor oe ertically. On either side of body are thin rods E 1 and E 2. E 1 and E 2 hae no angular elocity and hae a linear elocity as indicated. E 1 and E 2 contact in the sae anner as the configuration of figure 4 (although the fraes of reference for E 1 and E 2 are rotated by 90 with respect to figure 4). In figure 4, inconsistency occurred if external forces accelerated towards or ice ersa. The sae holds true for figure 5. If has an acceleration leftwards E 1 E 2 Figure 5. is constrained by the fixed wedges and can only oe horizontally. Howeer, the configuration is consistent only if is not subject to a net horizontal force. (towards E 1 ), then inconsistency occurs. Likewise, if has an acceleration rightwards (towards E 2 ), then inconsistency also occurs. Thus, the configuration of figure 5 is consistent only if the net horizontal acceleration of is zero. In this case, the rods E 1 and E 2 ski along the surface of as in figure 4b. Now consider figure 6, where a collection of thin rods R 1,...,Rn hae been added. In addition, an external horizontal force with agnitude µs acts on, trying to accelerate to the right. Each rod R i has ass i. The configuration between each rod R i and is the sae as the configuration of figure 3; thus each rod R i has angular elocity and is subject to an external graity force. Let f i be the agnitude of the noral force between R i and. s in figure 3, the only alid solutions for f i are f i = 0 and f i = i. If f i = 0, then no friction force acts between R i and. Otherwise, f i = i and a friction force of agnitude µ i acts between R i and. The friction force pushes R i to the right and to the left, with agnitude µ i. The friction force on therefore acts to oppose the external force of agnitude µs. R 1 R 2 R N µs... E 1 E 2 µf 1 µf 2 µf n Figure 6. The configuration is consistent if and only if the friction forces on su to µs. In order for the configuration of figure 6 to be consistent, ust hae no net horizontal acceleration. This eans that the friction forces exerted on fro the n rods ust su to µs, balancing the external force applied to. Thus, the configuration is consistent if and only if Σ n µf i =µs. (7) i = 1 Since each f i is either 0 or i, the configuration is consistent if and only if soe subset of { 1,...,n } sus to S. We can now perfor the reduction fro subset su to show NP-hardness. Gien any set = {a 1,...,an } and any target su S, construct the configuration of figure 6. ssign i = a i for 1 i n, and let an external horizontal force of µs act on as shown in figure 6. y the aboe discussion, the configuration is consistent if and only if there exists a subset of { 1,...,n } that sus to S. ut since = { 1,...,n }, the configuration is consistent if and only if (,S) is satisfiable. We conclude that the proble of deciding frictional consistency is NP-hard. 34

5 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 Theore 3. Deciding frictional consistency is NP-coplete. Proof. The result follows iediately fro Theore 1 and Theore 2. Corollary 1. Coputing contact forces (if they exist) for a configuration is NP-hard. Proof. Since deciding if a set of contact forces exists is an NPcoplete proble, coputing the contact forces (if they exist) is an NP-hard proble. 5.1 Iplications t this point, it ay see that the aboe results, while possibly of soe (arginal) theoretical interest, hae no bearing on any practical proble. Certainly, the aboe configurations were carefully constructed to produce configurations whose consistency was difficult to deterine. ut how likely is it that a configuration this carefully constructed could occur during siulation? For that atter, suppose the occurrence of any inconsistent configuration is so unlikely that the possibility can be copletely disregarded. (This ay be a reasonable assuption. We hae not encountered an inconsistent configuration during siulation when µ < 1.) Can a polynoial tie algorith that coputes contact forces only for consistent configurations be constructed? The answer to this is no, unless it turns out that P and NP are equialent, and it is widely belieed that they are not. Corollary 2. polynoial tie algorith for coputing alid contact forces for consistent configurations exists if and only if P = NP. Proof. Suppose that P = NP. Since quadratic prograing lies in NP, P = NP iplies a polynoial tie algorith for finding the solution to a QP. Since alid contact forces for a consistent configuration of bodies can be found by soling an associated QP, alid contact forces are coputable in polynoial tie if P = NP. Conersely, suppose that contact forces for consistent configurations can be coputed in polynoial tie. Then there exists a achine M and a polynoial p with the following behaior. Wheneer M is gien a consistent configuration C as input, M outputs a alid set of contact forces within tie p( C ). M s behaior when C is inconsistent is undefined. Gien any configuration C, not necessarily consistent, M can be used to decide consistency in polynoial tie as follows. Let C be input to M and run for p( C ) tie. If M fails to output within this tie, then C is inconsistent. Otherwise, M has produced soe output. Since deciding frictional consistency is in NP, the alidity of M s output can be decided in an additional aount of tie that is also a polynoial function of C. If M s output is a alid set of contact forces, then clearly C is consistent. If M s output is inalid, then C ust be inconsistent (else M would hae output a alid answer). In any eent, the consistency of C has been decided in polynoial tie. Since deciding consistency is NP-coplete, we conclude that the existence of a polynoial tie algorith for coputing contact forces on consistent configurations would iply that P = NP. Gien the aboe conclusions, it is unlikely that an efficient algorith for coputing contact forces can be found. This depressing result can be iewed in seeral ways. First, the siulation of rigid bodies with friction can be considered an intractable proble, unless the nuber of contact points with friction in a configuration is sall. Second, the general siulation proble can be rejected as being too difficult a proble, although we ight hope to find soe natural class of configurations with friction for which contact forces can be coputed efficiently. Such a class would hae to be sufficiently general to coer situations likely to be encountered in practice. Third, heuristic ethods for coputing contact forces can be considered. Howeer, this is essentially the sae as hoping to find a natural class of configurations with easily coputed contact forces. Rather than adopt any of these iewpoints, the next section presents a physical odel of inconsistency that leads to a natural reforulation of the proble of coputing contact forces. 6. Physical Models In this section, a physical odel for both inconsistency and indeterinacy is presented. Certainly, other odels are possible, and a different choice of odel ight lead to different conclusions and results. The odel in this section was deeloped in order to understand the behaior of inconsistent and indeterinate configurations. fter the odel was deeloped, we found that the odel leads to a natural refutation of the principle of constraints. y abandoning this principle, the proble of coputing contact forces is naturally reforulated and a correspondingly efficient way of coputing contact forces is found. The odel in this section is not an ad hoc attept at dealing with friction. We feel that the odel is not unreasonably based on the physical properties of rigid bodies, and sensible in the context of siulating rigid bodies with friction. The odel and subsequent reforulation of the proble is presented in this section. In the next section, a coputational algorith is presented for soling the reforulated proble. The otiation of a physical odel stes fro the need to answer the following basic question: what should be the result of a siulation when inconsistency is encountered? For inconsistent configurations, such as figure 4a, the only resolution is the introduction of an ipulsie contact force at p a [9, 13]. Ipulses, howeer, arise fro collisions between bodies. Gien the fact that p a has no elocity noral to (so that and do not appear to be colliding), why should an ipulse be applied between and? We answer this by presenting our physical odel of inconsistency. The physical odel we present is based on questioning the rigid body assuption. In the physical world, there is of course no such thing as a perfectly rigid body. For near rigid bodies, contact forces arise as a result of sall elastic deforations in the neighborhood of the contact area. Rather than geoetrically odel deforations, we shall (conceptually) allow bodies to inter-penetrate slightly, and consider a deforation in the contact surfaces proportional to the aount of inter-penetration. (We do not of course iagine that real bodies actually inter-penetrate). s the inter-penetration depth increases, a restoring noral force acts to oppose the inter-penetration. This is the so called "penalty ethod", a siulation ethod that odels contact between bodies as spring and daper systes. The noral force between two bodies is zero when the aount of inter-penetration is zero, and increases onotonically as the inter-penetration increases. Typically, the noral force is odeled as a linear spring force Kd, where K is the spring constant and d is the aount of interpenetration. lthough this is a ery useful conceptual odel, it is not well suited to siulation of ery rigid bodies[1, 3]. We will use the penalty ethod to conceptually odel inconsistency and indeterinacy, but we will not use the penalty ethod as a siulation technique. 6.1 Model of Inconsistency Figure 7 shows the behaior of the inconsistent configuration of 4a when the penalty ethod is applied. t tie t 0, consider the tip of the rod, p a, to be resting exactly on, with zero inter-penetration. Since there is no inter-penetration, the noral force is zero. Een though p a is sliding along, the friction 35

6 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 Tie t 0 : zero inter-penetration Tie t 0 + t: d 1 inter-penetration Tie t 0 : zero inter-penetration Tie t 0 + t:.a.a p a.a Kd 1 (nˆ +µtˆ ).a Figure 7. t tie t 0, only graity acts on. t tie t 0 + t, the inter-penetration distance is d 1 and both a penalty and a graity force act on, causing p a s downwards acceleration to increase. force is zero since the noral force is zero. Since the only force acting on is the external graity force, p a accelerates downwards. t tie t 0 + t, p a has inter-penetrated by an aount d 1, so a noral force Kd 1 nˆ acts on. Since p a is still sliding, a friction force of µkd 1 tˆ also acts on. The net result, fro equation (2), is that this causes p a to accelerate downwards een faster than before. s the penalty force continues to increase, it causes ore inter-penetration between and ; a for of positie feedback. ccordingly, both the friction and the noral force increase, and the cycle continues. Since we are trying to odel and as rigid bodies, the spring constant K ust be allowed to be arbitrarily large. (It is this feature that akes the penalty ethod ill-suited to rigid body siulation). The larger K is, the faster inter-penetration increases and the faster the noral and friction forces build. Recall that the friction force opposes the sliding otion of across. y aking K arbitrarily large, the friction force brings p a to rest (horizontally) in an arbitrarily short tie. Now, suppose K is adjusted so that p a coes to rest within tie t. Then the aount of inter-penetration is O( t 2 ), since the ertical distance traeled by p a depends quadratically on the tie for which it traels. In the liit as K goes to infinity, the contact force on acts as an ipulse and instantaneously brings p a to rest horizontally, without inter-penetration occurring. This ipulse also causes p a to acquire a noral elocity towards, bringing the into colliding contact. The (second) ipulse resulting fro this colliding contact can be coputed according to [16]. Once p a is at rest horizontally, dynaic friction is replaced by static friction. The Coulob friction odel states that the agnitude f static of static friction satisfies f static µf whereas f dynaic =µffor dynaic friction. (ctually, µ is typically larger for static friction than dynaic friction, but this has no bearing on the odel being deeloped.) ecause static friction is less constrained than dynaic friction, once static friction occurs, a alid solution exists and the inconsistency is reoed. 6.2 Model of Indeterinacy Consider the indeterinate configuration of figure 3, which has solutions f = 0 and f =. Using the penalty ethod, the indeterinacy can be reoed by assuing soe aount of initial inter-penetration between and. If the initial interpenetration between and is zero (figure 8) then no noral p a Figure 8. The initial inter-penetration is zero and only graity acts on. The centripetal acceleration of pulls p a away fro and contact is broken. force exists, and contact is iediately broken (due to the centripetal acceleration of p a away fro ). The behaior is the sae as in figure 3a. Howeer, if the initial inter-penetration produces a noral force agnitude of, then the noral and friction forces preent fro breaking contact with. In figure 9, let the initial inter-penetration d 1 be. K Tie t 0 : d 1 inter-penetration Kd 1 (nˆ +µtˆ ).a = 0 Tie t 0 + t: d 2 d 1 inter-penetration Kd 2 (nˆ +µtˆ).a = 0 Figure 9. The initial inter-penetration is d 1. oth graity and a penalty force act on. slides and falls without breaking contact with. Then the noral force agnitude at tie t 0 is. Since p a is sliding on, a friction force acts on as shown. s falls, aintaining contact with, the inter-penetration aries soothly, produce a arying noral force. t tie t 0 + t, still interpenetrates by an aount d 1 d 2, and the behaior of the configuration is that of figure 3b. Thus, the initial aount of inter-penetration deterines which behaior occurs. The siulation ethod of coputing and applying contact forces and ipulses to bodies does not odel inter-penetration. Instead of deterining behaior by initial choice of interpenetration, we can consider an initial noral force between bodies at contact points, and use that to deterine subsequent behaior. For the applications we are interested in, we generally hae no basis for preferring one set of initial noral forces oer another. The nuerical routines used for soling the contact force equations arbitrarily deterine the behaior siulated. This ay or ay not be sensible for other applications. 36

7 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July The Principle of Constraints The principle of constraints, applied to configurations with friction, states the following: when coputing forces for a configuration of bodies, ipulsie forces should be used only if non-ipulsie forces do not exist for the configuration. In other words, if a configuration is consistent, non-ipulsie forces should be coputed and applied to the configuration; otherwise ipulsie forces ust be introduced into the syste. Initially, this sees like a sensible principle, but we know of no real justification for it. If the physical odel presented in this section is adopted, then this principle ust be abandoned (at least in the context of rigid body siulation). Consider figure 10. Once again, the cobination of graity and the angular elocity of R 1 is the sae as in figures 5 and 6. Siilarly, a horizontal acceleration of results in inconsistency. E 1 µf 1 Figure 10. f 1 ust be either 0 or 1 to be alid. Howeer, f 1 = 1 causes inconsistency. The only alid solution is f 1 = 0. If E 1 is ignored for the oent, then both f 1 = 0 and f 1 = 1 are alid solutions for f 1. The only alid solution for the configuration as a whole though, is f 1 = 0; f 1 = 1 pushes to the left, causing inconsistency. Howeer, using the physical odel of indeterinacy, the alue f 1 assues depends on the initial inter-penetration between R 1 and. If we adopt the physical odel presented, we ust conclude the following: een though the configuration is consistent, there is no a priori reason to prefer ipulse-free behaior to non-ipulse-free behaior for this configuration. The inconsistency resulting fro f 1 = 1, and subsequent application of ipulsie contact forces is as acceptable a behaior as the application of non-ipulsie contact forces resulting fro f 1 = 0. Een though the configuration in figure 10 has only one alid solution of contact forces, (f 1 = 0), it has two possible behaiors and is thus indeterinate. 6.4 Reforulating the Contact Force Proble Up to now, we hae iewed the proble of coputing alid contact forces as: gien a configuration, efficiently copute a alid set of contact forces, if they exist. This iewpoint is based on the principle of constraints; that is, ipulsie forces should be applied if and only if the configuration is inconsistent. It is this absolute insistence on a non-ipulsie solution, if it exists, that akes the proble of coputing contact forces so difficult. Howeer, now that we hae abandoned the principle of constraints, a different iewpoint of the proble is possible. We reforulate the proble of coputing contact forces as: gien a configuration, efficiently copute either a alid set of contact forces or a alid set of contact ipulses. (Validity for contact ipulses is defined in section 7). Under the physical odel we hae assued, there is no intrinsic reason to prefer alid contact forces oer alid contact ipulses. y coputing a particular set of alid contact forces or ipulses, a particular behaior is chosen for the configuration, and other possible behaiors ignored. This eans that we do not R 1 bother to decide if a configuration is consistent or not. If a alid set of contact ipulses are coputed, it will not be known if the configuration was consistent and could hae been soled with contact forces; howeer, this is uniportant. In the next section, an efficient ethod is presented for coputing alid contact forces or ipulses. 7. Coputing Valid Contact Forces and Ipulses efore an efficient ethod for coputing either contact forces or ipulses can be considered, the definition of alidity ust be extended to coer contact ipulses. We first define alidity for contact ipulses and then present a coputational algorith. 7.1 Valid Contact Ipulses In the penalty ethod interpretation of figure 7, an ipulse occurred because no atter how strong the noral force becae, it was insufficient to preent inter-penetration. s a result, after the contact ipulse was applied, the relatie elocity of the bodies at the contact point was directed inwards. Since contact ipulses ay need to be applied to configurations inoling ore than one contact point, alidity ust be defined for a set of contact ipulses. For exaple, in figure 10, if the f 1 = 1 behaior is chosen, a contact ipulse should occur between E 1 and. Howeer, there should be no contact ipulse between R 1 and. In order for our definition of alidity to be useful, all inconsistent configurations should hae a alid set of contact ipulses. We show in section 7.2 that our definition of alidity for contact ipulses satisfies this requireent. We call a set of contact ipulses alid under the following two conditions. First, the contact ipulses ust conert at least one of the contact points with dynaic friction to static friction. Second, eery contact point at which a contact ipulse occurs ust end up with a non-positie relatie noral elocity; that is, after the contact ipulses are applied, bodies should not be separating whereer contact ipulses occurred. The justification for this is that the contact ipulses occur only when the noral force grows without bound to oppose inter-penetration. Intuitiely, alid contact ipulses are the liiting result of increasing noral forces without bound under the penalty ethod. If bodies are separating at a contact point after contact ipulses are applied, then the noral force at the contact point should not hae grown without bound into a contact ipulse. s in section 6.1, bodies will be colliding at soe contact points after alid contact ipulses are applied, and a secondary set of ipulses will hae to be applied. These ipulses ay be calculated according to [16]. 7.2 Coputing Contact Forces and Ipulses with Leke s lgorith How can either contact forces or ipulses be coputed efficiently, gien that coputing contact forces alone is hard? In section 3, it was stated that eery configuration of n contact points had an associated quadratic prograing proble of n ariables. Let a set of alid noral force agnitudes (if it exists) be denoted by the unknown n-ector f ; the agnitude of the ith noral force is gien by f i.if f exists, it can be found by soling the QP iniize T f (f +b f 0 ) subject to f f +b 0 (8) where and b are deterined by the configuration. is an n n inerse ass atrix and b is an n-ector of known external and inertial accelerations. f + b represents the relatie accelerations at contact points. (See [1, 2, 6, 12] for a discussion of the nuerical properties of and ethods for coputing ). Eery 37

8 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 f such that equation (8) attains zero is alid. If equation (8) cannot attain zero subject to the aboe restrictions, then the configuration has no alid solution and is inconsistent. Thus, a alid f is a solution to the equation f T (f +b )=0, f 0 and f +b 0. (9) Equation (9) is what is known as a linear copleentarity (LCP) proble. Equation (9) is called a positie seidefinite (PSD) LCP if is PSD[14]. One of the first algoriths for soling linear copleentarity probles was introduced by Leke[10] and is known as Leke s algorith. Leke s algorith is a pioting ethod, siilar to the siplex ethod of linear prograing and has siilar nuerical properties. The algorith is exponential in the worst case, but has an expected running tie polynoial in n[14]. Leke s algorith progresses, like the siplex ethod, by trying arious descent directions. If an LCP is PSD and has no solution then Leke s algorith will at soe point encounter an unbounded ray ; a descent direction along which one can trael infinitely far without aking any progress. Otherwise, if a PSD LCP has a solution, then no unbounded ray exists for that LCP, and Leke s algorith terinates by finding a solution to the LCP. The algorith is iewed as a practical solution ethod to the proble of soling PSD LCP s. Howeer, for non-psd LCP s, Leke s algorith is not guaranteed to terinate correctly (although it still takes only expected polynoial tie to do so). For a non-psd LCP, if there is no solution, Leke s algorith terinates by encountering an unbounded ray. Unfortunately, if there is a solution, the algorith is not guaranteed to find it. For non-psd LCP s with solutions, Leke s algorith terinates either by finding a solution or by encountering an unbounded ray. 1 s a result, Leke s algorith is not suitable for soling non-psd LCP s. Howeer, when Leke s algorith terinates by encountering an unbounded ray, it has found an n-ector z with the property[14] z 0 and i such that z i > 0, (z ) i 0 (10) where (z ) i is the ith coponent of the ector z. Why is this property of interest? Suppose that a set of contact ipulses are applied to the configuration, with the agnitude of the noral ipulse at the ith contact point denoted by z i. Then it can be shown[2, 6] that the relatie elocity at the ith contact point after the ipulse is (z ) i. If the ector z satisfies equation (10) then eery contact point subject to a non-zero contact ipulse z i > 0 ends up with a non-positie relatie noral elocity (z ) i 0. Thus, the ector z found by Leke s algorith gies rise to a alid set of contact ipulses. To fully satisfy the definition of alidity, z ust be scaled upwards fro zero until it causes a contact point with dynaic friction to be conerted to static friction. fter this, a real ipact occurs, as described in section 6.1. The behaior of Leke s algorith exactly atches our new iew of the proble of coputing contact forces. If the configuration has no alid contact ipulse solutions, Leke s algorith cannot terinate with the special ector z and ust therefore find a alid contact force solution. For inconsistent configurations, no alid contact force solution exists, so Leke s algorith ust terinate with the ector z, proiding a contact ipulse solution. For configurations with both a alid force and ipulse solution, Leke s algorith will terinate by coputing 1 Encountering an unbounded ray when there is a solution is analogous to getting stuck at a non-global iniu in a non-conex iniization proble. one or the other. Wheneer Leke s algorith terinates by coputing a contact ipulse solution, it will still be unknown whether or not the configuration was consistent. For frictionless systes, the LCP is always PSD and has a solution, so frictionless configurations do not hae alid ipulse solutions. Thus, the reforulation of the proble does not add any new solutions to siulations of frictionless systes. lthough Leke s algorith runs, practically speaking, in polynoial tie, this is not a proof that finding either alid contact forces or ipulses is a polynoial tie proble. Fro a practical standpoint, though, Leke s algorith proides an efficient algorith for coputing alid contact forces or ipulses. The coputational coplexity of either soling an LCP or finding an unbounded ray is unknown. 8. pproaches for Static Friction We conclude with two approaches to dealing with static friction. We stress that these approaches are only a first step towards dealing with the probles of static friction. oth approaches hae their drawbacks, and currently hae only liited applicability. The two approaches appear to produce (approxiately the sae) reasonably realistic results for the configurations we hae siulated. Consider the ith contact point of a configuration, and let the noral force agnitude there be f i. The coefficient of friction, µ, is not indexed and ay be different for each contact point. No distinction is ade between the coefficient of static and dynaic friction, and both are assued to be isotropic. In what follows, there is no difficulty in using a different alue of µ depending on whether the friction force is static or dynaic. The next few coputations take place in the tangent plane of the contact surface at each contact point; ectors are expressed in this plane as pairs (x,y) where (1,0) and (0,1) are orthonoral. Let (f xi, f yi ) be the friction force, and ( xi, yi ) and (a xi,a yi ) the relatie tangential elocity and acceleration between bodies at the ith contact point. 2 If ( xi, yi ) is non-zero, then dynaic friction occurs and and the friction force has agnitude µf i and is anti-parallel to the ector ( xi, yi ). Static friction is ore coplex. For static friction, (f xi, f yi ) 2 = f 2 xi + f 2 yi (µf i ) 2. (11) The ain difficulty in static friction is deterining when a contact points akes a transition fro sticking to sliding. When the static friction force is sufficient to preent sliding, any direction of the friction force constraining (a xi,a yi ) to be zero is alid. If the body begins to slide, then (f xi, f yi ) ust at least partially oppose the acceleration; that is, (f xi, f yi ). (a xi,a yi ) 0. (12) lso, if (a xi,a yi ) is non-zero, then the friction force agnitude ust attain its upper bound of µf i. The law for static friction can be suarized as f 2 xi + f 2 yi (µf i ) 2,(f xi,f yi ). (a xi,a yi ) 0 and ((µf i ) 2 (f xi 2 + f yi 2 ))(a xi 2 + a yi 2 ) = 0 (13) where the last condition forces either (µf i ) 2 = f 2 2 xi + f yi or a 2 xi + a 2 yi = 0. Unfortunately, equation (13) is too coplex to be 2 Care ust be taken here. The relatie acceleration (a xi,a yi ) is calculated by taking the first deriatie of a elocity constraint, not the second deriatie of a spatial constraint. See Goyal[8] for details. 38

9 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 forulated as part of a quadratic progra. It also does not appear practical to sole with current non-linear prograing techniques. 8.1 The Dynaic Friction pproxiation This approach for approxiating static friction is extreely siple to ipleent. In order to deterine whether static friction or dynaic friction should occur at a contact point, a siulator ust hae soe threshold alue ε. If ( xi, yi ) ε, then dynaic friction occurs. Otherwise ( xi, yi ) < ε and static friction occurs. Since dynaic friction is ( xi, yi ) (f xi, f yi ) =µf i (14) ( xi, yi ) we approxiate static friction as ( xi, yi ) ( (f xi, f yi ) = xi, yi ) ( µfi xi, yi )µf =. i (15) ε (, xi yi ) ε Thus, we really use a dynaic friction force that aries in agnitude fro zero to an upper liit of µf i as the relatie contact speed aries fro 0 to ε. This allows us to use the ethod of section 7.2 to copute both static and dynaic friction. Since static friction occurs only when the relatie tangential elocity is non-zero, bodies ust acquire soe sall aount of "crawl" in order to aintain a static friction force. This approach is reiniscent of the penalty ethod, where bodies ust acquire soe degree of inter-penetration for a sufficient noral force to exist. Howeer, in the penalty ethod, it is necessary to increase the spring constant K without bound as the ass of bodies increases. Our approxiation ethod does not suffer fro this proble. If ε is ade sall enough, the "crawling" behaior of bodies is not isible, no atter what asses or forces exist. If ε is ade excessiely sall, the differential equations of otion[1, 3] ay becoe stiff; otherwise, the approach has a reasonable perforance. The ajor adantage to this approach is that it is guaranteed to produce a result, using Leke s algorith as described in section 7.2. Thus, either a set of contact forces or ipulses is coputed. The ajor disadantage to this approach is that it is an ad hoc approxiation to the law of static friction. 8.2 Modeling Static Friction by Quadratic Prograing This approach is uch ore abitious. We attept to odel static friction as a quadratic prograing proble, which can be soled to find the contact forces. We approxiate the static friction law as follows. Equation (11) is rewritten as µf i f xi µf i and µf i f yi µf i. (16) Unfortunately, this allows the static friction force agnitude to exceed µf i (by as uch as a factor of ), unless the friction force happens to be aligned with a coordinate axis of the tangent plane. One possible solution is to iterate seeral ties, trying to choose a coordinate syste so either f xi or f yi is zero, for each contact point. For two-diensional configurations howeer, the friction force is constrained to a line, not a plane, and is described by a single ariable f xi. In this case, the constraint µf i f xi µf i is exact. To satisfy equation (12), we add the conditions f xi sgn(f xi ) 0 and a xi sgn(f xi ) 0 f yi sgn(f yi ) 0 and a yi sgn(f yi ) 0 where sgn(x) = 1ifx 0 and 1 otherwise. (17) These conditions ensure that (f xi, f yi ). (a xi,a yi ) 0. The condition that static friction attains its upper bound when slipping begins is written (µf i f xi sgn(f xi ))(a xi sgn(f xi )) = 0 (µf i f yi sgn(f yi ))(a yi sgn(f yi )) = 0. (18) Finally, we add the standard constraint on the noral forces that f i 0, a i 0 and f i a i = 0 (19) where a i is the relatie noral acceleration of the ith contact point. If the signs of the f xi and f yi are known, then the aboe syste of equations has unknown ariables f i, f xi, f yi (for each contact point) which are used to express the a i, a xi and a yi ters. The entire syste can be soled by a quadratic progra because the sgn functions becoe known. How can the signs of the f xi and f yi ariables be deterined? Iteratie ethods for quadratic prograing and linear copleentarity exist that can be adopted to this proble[14]. These iteratie techniques are ery siilar to the Gauss-Seidel or Jacobi iteratie ethods used to sole linear systes. Iteratie ethods for quadratic prograing are odified in a straightforward fashion to sole the syste of equations (16) thru (19), without initially knowing the signs of the f xi and f yi. Unfortunately, conergence results are not aailable for the odified iteratie ethods. If the odified ethod fails to conerge (or een before full conergence), the signs of f xi and f yi can be guessed by exaining the unconerged solution. Quadratic prograing is then used to sole equation (16) thru (19) as a quadratic progra, gien the estiate of the signs of the ariables. If the estiate is correct, a solution is obtained for the friction forces. Howeer, the approach can break down at any nuber of places. If the ethod fails to conerge, the estiates of the signs of the ariables ay not be correct. Een if the signs of the ariables are correct, the for of the linear constraints in equation (16) do not allow us to use Leke s algorith for linear copleentarity. lthough we can apply standard quadratic prograing ethods, we know of no algorith that will sole the quadratic progra or indicate contact ipulses, as Leke s algorith does. With regard to the entire issue of consistency and NP-hardness, this ethod for static friction is back to square one. It is possible that when the iteratie step fails to conerge, an analysis of the diergence of the iterates will indicate a alid set of contact ipulses. t this tie, howeer, we do not know how to perfor such an analysis. We hae found howeer that the second approach, when it works, yields a ery acceptable result. For large nubers of contact points (n 40), the second approach soeties breaks down, while the first approach does not. We hae had reasonable success with the second approach for configurations with 40 contact points or less. 9. Conclusion n efficient algorith for dealing with configurations of bodies with only dynaic friction has been presented. Instead of attepting to force a behaior that aoids contact ipulses, the algorith allows either contact forces or contact ipulses to occur. Two preliinary approaches for dealing with static friction are presented. The first approach is an approxiation using the algorith deeloped for siulating dynaic friction. The second approach is ore exact but also ore prone to failure than the first approach. Siulation of a coplex configuration with static and dynaic friction is shown in figure

10 SIGGRPH 91, Las Vegas Coputer Graphics, Volue 25, Nuber 4, July 1991 cknowledgeents This research was funded by an T&T ell Laboratories PhD Fellowship and two NSF grants (#DCR and #SC ). Siulations were perfored on equipent generously donated by the Hewlett Packard Corporation and the Digital Equipent Corporation. I wish to thank both ruce Donald and ndy Ruina for seeral stiulating conersations about NPcopleteness results and frictional behaior. ppendix : cceleration due to Contact Force We copute the noral acceleration of p a,.a, for the configuration of figure 2. Let a and α denote the linear and angular acceleration of,, the angular elocity of, and r, the displaceent of p a fro the center of ass of. Vectors are treated as 3-space ectors: a, p a, ṗ. a, nˆ and r lie in the xy plane while and α are parallel to the z axis. Fro figure 1, r = ( cosθ, sinθ,0). ṗ. a ay be expressed as the su of three ters: the linear acceleration a, the tangential acceleration α r, and the centripetal acceleration ( r ). The linear acceleration, a,is a = fnˆ +µftˆ + ( g)nˆ = fnˆ +µftˆ gnˆ. (20) The torque on about its center of ass is r (fnˆ +µftˆ), which yields an angular acceleration of α r = (fnˆ +µftˆ). (21) I Then ṗ. a = a +α r + ( r ) fnˆ +µftˆ = gnˆ + r (fnˆ +µftˆ) r + ( r ). (22) I Taking the dot product of equation (20) with nˆ, nˆ. a fnˆ. nˆ +µfnˆ. = tˆ gnˆ. nˆ = f g. (23) Taking the dot product of the tangential acceleration α r with nˆ and using the geoetry of figure 1, nˆ. ( r ) = 2 sinθ and nˆ. (α r ) = nˆ. r (fnˆ +µftˆ) r (24) I 3. arzel, R. and arr,.h., odeling syste based on dynaic constraints, Coputer Graphics (Proc. SIG- GRPH), ol. 22, pp , Cottle, R.W., On a proble in linear inequalities, Journal of the London Matheatical Society, ol. 43, pp , Erdann, M.., On Motion Planning with Uncertainty, M.S. Thesis, Massachusetts Institute of Technology, Featherstone, R., Robot Dynaics lgoriths, Kluwer, oston, Garey, M.R. and Johnson, D.S., Coputers and Intractability, Freean, New York, Goyal, S., Second order kineatic constraint between two bodies rolling, twisting and slipping against each other while aintaining point contact, Technical Report TR , Departent of Coputer Science, Cornell Uniersity, Kilister, W. and Reee, J.E., Rational Mechanics, Longan s, London, Leke, C.E., iatrix equilibriu points and atheatical prograing, Manageent Science, ol. 11, pp , Lötstedt, P., Coulob friction in two-diensional rigid body systes, Zeitschrift für ngewandte Matheatik un Mechanik, ol. 61, pp , Lötstedt, P., Nuerical siulation of tie-dependent contact friction probles in rigid body echanics, SIM Journal of Scientific Statistical Coputing, ol. 5, no. 2, pp , Mason, M.T. and Wang, Y., On the inconsistency of rigidbody frictional planar echanics, IEEE International Conference on Robotics and utoation, Murty, K.G., Linear Copleentarity, Linear and Nonlinear Prograing, Helderann Verlag, erlin, Vaasis, S.., Quadratic Prograing is in NP, Technical Report TR , Departent of Coputer Science, Cornell Uniersity, Wang, Y. and Mason, M.T., Two diensional rigid body collisions with friction, Journal of pplied Mechanics, (to appear). = f(cos 2 θ µcosθsinθ). I Then, fro the relations in figure 1, nˆ.. ṗ a = f g + f(cos 2 θ µcosθsinθ) + 2 sinθ I = f (1 + 16(cos 2 θ µcosθsinθ)) + 2 sinθ g (25) = f (1 2) + 2 sinθ g= f + ( 2 sinθ g). References 1. araff, D., nalytical ethods for dynaic siulation of non-penetrating rigid bodies, Coputer Graphics (Proc. SIGGRPH), ol. 23, pp , araff, D., Cured surfaces and coherence for nonpenetrating rigid body siulation, Coputer Graphics (Proc. SIGGRPH), ol. 24, pp , Figure 11. Siulation of a coplex configuration. 40