Modeling Complex Contacts Involving Deformable Objects for Haptic and Graphic Rendering

Transcription

1 Robotics: Science an Systems 2005 Cambrige, MA, USA, June 8-11, 2005 Moeling Complex Contacts Involving Deformable Objects for Haptic an Graphic Renering Qi Luo IMI Lab, Department of Computer Science University of North Carolina - Charlotte Charlotte, NC 28223, USA qluo@uncc.eu Jing Xiao IMI Lab, Department of Computer Science University of North Carolina - Charlotte Charlotte, NC 28223, USA xiao@uncc.eu Abstract Haptic renering involving eformable objects has seen many applications, from surgical simulation an training, to virtual prototyping, to teleoperation, etc. High quality renering emans both physical fielity an real-time performance, which are often conflicting requirements. In this paper, we simulate contact force between a hel rigi boy an an elastic object an the corresponing shape eformation of the elastic object efficiently an realistically base on a nonlinear physical moel an a novel beam-skeleton moel, taking into account friction, compliant motion, an multiple contact regions. Our approach is able to achieve a combine upate rate of over 1 khz in realistic, smooth, an stable renering, as emonstrate by our implemente examples. I. INTRODUCTION Moeling eformable objects in contact has been stuie both for graphic renering an for haptic renering. Early research work on graphic renering of eformable objects was surveye by Gibson an Mirtich [1]. More recent survey on graphic an haptic renering involving eformable objects can be foun in [2]. The existing work can be ivie mainly into two large categories of approaches: purely geometric approaches (incluing methos base on splines, patches an free-form geometric moels) an physically base approaches (base on mass-spring moels an continuum moels). While graphic renering only nees to make the moele object eformation look realistic, haptic renering requires that the eforme object feels realistic as well, an, comparing to visual images, humans ten to have a more acute, finer, an richer tactile sense of object properties. While the upate rate in graphic renering nees to be aroun Hz to look realistic, the upate rate in haptic renering nees to reach 1 khz to feel realistic. Therefore, haptic renering requires much higher level of physical realism an much faster upate rate than graphic renering to achieve high quality an realistic renering, which is essential for many applications that simulate manipulations in real physical worl. However, high-level of physical realism an fast upate rate of renering are often conflicting requirements. A common approach is to aopt the physically-base eformable moels use in graphic renering, such as mass-spring-amper moels an continuum moels, an to improve the upate rate by applying certain simplifications to spee uomputation. Mass-spring-amper moels are rather popular in haptic renering (e.g. [3][4]). Such moels are simple with well unerstoo ynamics, easy to construct an can be use for interactive an even real-time simulation. However, it has rawbacks. The physical accuracy of such a moel is often not sufficient. For example, incompressible volumetric objects or thin surfaces that are resistant to bening are ifficult to be moele as mass-spring systems. The moel is linear, an in orer to simulate nonlinear force response, it is necessary to integrate the linear moel in a way similar to using Finite Element Metho (FEM), but then real-time upate rate for haptic renering is har to achieve. As for continuum moels, such as moels base on FEM [5], Finite Difference Metho [6], Bounary Element Metho [7], an Long Element Metho [8], real-time performances can be achieve only by further simplifications or aaptive methos. With the ai of pre-calculations an multiresolution methos, eformation of more complicate objects can be simulate in real-time. However, little is one to eal with eformable objects uner complex contact states involving multiple contacts an compliant motions with friction, which coul make real-time performance har to achieve. The majority of previous approaches have assume single contact region an localize eformation [9]. Increasing renering spee is always at the expense of lowering physical accuracy. In this paper, we simulate contact force between a hel rigi boy an an elastic object an the corresponing global shape eformation of the elastic object efficiently an realistically base on a nonlinear physical moel an a novel beamskeleton moel. Our approach takes into account friction, compliant motion, an multiple contact regions. The objective is both to avoi expensive computation an to preserve physical accuracy of continuum moel. We use the general Duffing equation [10] as a founation to simulate nonlinear contact forces from a eforme object in contact. The Duffing equation is one of the stanar moels for stuying nonlinear systems subjecte to external forces. It is well stuie, relatively simple, an yet is powerful to moel very complex behaviors [11]. This moel is particularly suitable for moeling the nonlinear stiffness of biomaterials, as in surgical simulations. Contact forces of ifferent types of eformable objects (i.e. elastic, plastic, etc) can be simulate

2 by changing the relate parameters, which can be achieve by pre-calculations [9]. We introuce a novel beam-skeleton moel to compute the stresses an strains of a eforme elastic object at certain extremal points as well as the stresses at multiple contact regions, base on which, we further introuce fast computation of global shape change through an interpolation metho that achieves minimization of elastic energy. Moreover, we take into account the ifferent effects of ifferent contact areas on shape change (uner the same force). The rest of the paper is organize as follows. In Section II, we introuce some basic assumptions. In Section III, we briefly escribe real-time collision etection use in our approach. In Section IV, V, an VI, we escribe our metho for contact force moeling an beam-skeleton moel for graphic renering of shape eformation an for ealing with multiple contact region cases. We present some implementation results in Section VII an conclue the paper in Section VIII. II. BASIC ASSUMPTIONS Homogeneous Isotropic Elastic Material Depening on material properties, eformable objects can be categorize into many types. In this paper, we focus on moeling objects that are mae of homogeneous isotropic elastic material. The overall eformation effect for such material is nonlinear. However, the nonlinear eformation only exists insie a small neighborhoo of the contact point/region where the stress is very high. The stress felt in other regions is much less an can be consiere as linearly istribute. Stable Equilibrium Configurations We only consier moeling the contact forces cause by quasi-static collision an compliant motion, which means that motions are slow enough such that only eformation occurre at stable equilibrium configurations nees to be consiere, where the elastic energy is minimize. This provies an effective iscretization of the otherwise continuous force an shape change happene on the elastic object in contact. Objects an Contacts We use a mesh moel representation for the geometry of the rigi hel object. For the elastic object, we maintain both a mesh moel an an exact parametric surface moel of its uneforme shape (especially if it is non-polyheral). The exact parametric surface moel is use for both fast an accurate computation of shape eformation (Section V). We efine a single contact region as a cluster of contact points S such that the istance between a contact point in S an its nearest neighboring contact point in S is less than a threshol r. A contact point outsie S is consiere belonging to another contact region, an there can be multiple contact regions in general. We only focus on cases where each single contact region is relatively small so that within the contact region, the first partial erivatives of the originally uneforme surface of the elastic object harly change. A contact region may consist of just a single contact point. For any point on the elastic object outsie certain immeiate neighborhoo R of a contact region, we consier its eformation as cause by the stresses an strains sprea to it from the contact region as a function of the contact force an call it global eformation. For points insie the neighborhoo R of the contact region, we take into account that the shape eformation is not only cause by the contact force but also by the size of the contact region: the greater the size, the smaller the unit pressures are uner the same force, an thus the smaller the eformation. We therefore moify the shape eformation insie R accoringly an call the result local neighborhoo eformation (Fig. 1). (a) (b) (c) Fig. 1. Example of shape eformation: (a) originally uneforme elastic object, (b) global eformation over the whole elastic object (the ot line shows the surfaces before eformation), (c) local neighborhoo eformation Force from the Human User Here we assume that the force exerte to the hel object from the human user is applie to the mass center of the hel object. This assumption is useful later for estimating the istribution of contact pressure (see Section IV.C). III. REAL-TIME COLLISION DETECTION Collision etection for rigi objects has been well investigate. However, real-time collision etection involving a eforming object in interactive environment poses special challenges. Our strategy is to treat contact etection between a rigi boy an an elastic object as two rigi boies at the time of contact an then ynamically change the surface moel of the elastic objet to reflect the shape changes ue to contacts. Specifically we use a real-time collision etection package SOLID[12] for objects in mesh moels that allows ynamic upates of meshes. In each time step i, contact etection is characterize by one of the following three phases. In the first phase, no contact has yet happene between the two objects, an thus the elastic object oes not change its shape. Here collision etection can be consiere as just between two rigi objects (in mesh moels). The secon phase is marke by the transition from no contact to contact between the two objects, i.e., at least one contact point is establishe in the current time step i. Once that happens, at least one contact region is forme an recore. The thir phase escribes the situation when the two objects were alreay in contact in the previous time step i 1 an remain in contact in the current time step i, but the contact region(s) may change. There are two possible changes: (1) existing contact region changes because the contact points in it change, an (2) new contact regions are forme. Our contact etection algorithm eals with the situation by comparing the contact points etecte in time step i 1 with those etecte

3 in the current time step i to upate existing contact regions an/or form new contact regions (in terms of mesh moels). Note that collision etection can again be treate as between two rigi objects in this phase, but the uneforme shape of the elastic object is replace by the eforme shape obtaine from time step i 1. After all contact regions are etermine, contact forces an the corresponing shape change of the elastic object can be compute an renere haptically an graphically for time step i in real-time as etaile in Sections IV, V, an VI. IV. CONTACT FORCE MODELING FOR HAPTIC RENDERING In this section, we first apply the general Duffing equation to provie a nonlinear contact force moel for a single point contact cause by pressing the rigi object normally to a face of the elastic object (Fig. 2). Then we exten the metho to moel other single point contact or single region contact cases. D X Fig. 2. Contact force simulation: a single point contact with normal compression (the hollow arrow inicates contact force irection) A. General Duffing Equation One of the commonly use nonlinear equations for characterizing the behaviors of nonlinear mechanical, electrical an chemical systems is the Duffing equation. We can use the Duffing equation to characterize the nonlinear force response of an elastic object in the basic single point contact case at a quasi-static state as shown in Fig. 2, where p 0 inicates the position of the contact point before eformation, which we call the origin of the eformation istribution, an D is the istance from p 0 to the point of maximum eformation, which we call the eformation isplacement vector with magnitue D. The Duffing equation essentially efines a nonlinear spring-amper-restorer moel. A general Duffing equation has the following form: p 0 ẍ + 2µẋ + ω 2 0x + ǫβ 2 0x 3 = AcosΩt (1) where x is the eformation istance, ω 2 0x is the linear restoring term, 2µẋ is the amping term, ǫβ 2 0x 3 is the nonlinear restoring term (with ǫ << 1), AcosΩt is proportional to the external force, an Ω is a constant. Note that the nonlinear restoring force item represents the non-linear properties offere by the eformation of the nearby area. It is reasonable to assume that the force exerte to the hel rigi boy from the human operator is constant uring one short time step. By using the equivalent frequency ω [10], ω = ω 0 + 3β 2 0a 2 ǫ/8ω 0, where a is the value of x at steaystate, we have the solution as x = e µt C sin ω 2 µ 2 t + A ω 2 X X X D (a) (b) (c) Fig. 3. Contact force simulation: (a) contact is normal to the initial uneforme surface, (b) skewe eformation, (c) compliant motion with eformation where the first part in the above equation is the transient term an the secon part is the steay-state term. With the quasistatic assumption (Sect. II.B), uner a large µ, the steay-state is achieve when x reaches D at the en of one time step, i.e., D = a = A ω 2 (2) That is, at x = D = a, the hel rigi object becomes static, an the contact force response F c from the elastic object balances the external force F e = ma, where m is the mass of the rigi object. Thus, from (2), by omitting the high orer term of ǫ (since ǫ << 1), we have F c = m ω 2 0D + 3β2 0ǫ 4 D3 (3) B. Skewe Deformation an Compliant Motion Now we exten our basic metho of contact force simulation to point contact cases where the irection of eformation is not normal to the initially uneforme surface of the elastic object so that the hel rigi object may get stuck or perform compliant motion on the elastic object. When the irection of eformation is not normal to the originally uneforme contact surface of the elastic object (Fig. 3), the eformation isplacement vector D can be ecompose into tangential an normal components D t an D n respectively. The force response ue to eformation along the normal irection F cn can be compute from (3) with D = D n, pointing to the irection against D n. Now we nee to etect whether the rigi boy is stuck or performs a compliant motion tangentially along the contact surface of the elastic object. First, assume that the rigi boy is stuck at the current time step i ue to the tangential eformation force response from the elastic object F ct, which can be compute from (3) with D = D t, pointing to the irection against D t. Accoring to [13], the maximum friction from objects of ifferent elasticity is proportional to F cn β 2, 3 β 1, in the empirical equation f max = K F cn β where the coefficient K an β were given in [14] for various eformable materials. For a truly elastic soli, β = 2 3. Now if F ct f max, our assumption is correct: the rigi object is inee stuck by the friction so that it will not have a compliant motion at the current time step. Only a skewe eformation happens (see Fig. 3b. Note that the shape of eformation is moele in Section V). The total contact force response from the elastic object to the rigi boy is F c = F cn + F ct. Otherwise, if F ct > f max, this represents an impossible case for static friction, inicating that our assumption that the rigi boy is stuck is incorrect. On the contrary, the rigi D D

4 object in fact makes a compliant motion, which shifts p 0 (i.e., the origin of the eformation istribution) from the previous time step i 1 to the current time step i. If the contact point (on the hel rigi object) has move tangentially from time step i 1 to time step i with a istance, then to moel the effect of compliant motion, we also shift p 0 the istance to obtain its new position. For the ynamic friction F ct, we have F ct = µ D F cn when µ D F cn < f max ; Otherwise we have F ct = f max, where µ D is the ynamic friction coefficient. Subsequently, the total contact force response from the elastic object to the rigi boy is F c = F cn + F ct. Accoringly, there is a shift of the eforme shape of the elastic object at time step i from that at time step i 1 ue to compliant motion (as shown in Fig. 3c. See Sect. V for moeling of the shape of eformation). C. Single Region Contact A single region contact is forme by a contact region of more than one point. The total effect of contact forces can be obtaine by integrating contact forces generate on contact points (or infinitesimal contact areas) over the whole contact region. We iscretize the force integration as the summation of contact forces responing to a number of evenly istribute contact points with ifferent isplacements of eformation, as shown in Fig. 4. To achieve real-time processing, the iscretization can be simply base on the vertex points of the mesh moel of the contact region of the rigi object, provie that these vertex points are evenly istribute on the mesh. The contact force response F i at each contact point p i can be calculate by the general Duffing equation base on its eformation isplacement i (Fig. 4) an the mass m i istribute on it. Summing up all F i gives the total force F against the irection of eformation. i i (a) (b) Fig. 4. Examples of single region contact an contact region iscretization ( is the irection of eformation): (a) normal eformation (b) skewe eformation In the case of a normal eformation (Fig. 4a), the compute F is the total contact force response F c along the normal of the originally uneforme surface of the elastic object corresponing to the single region contact. However, in the case of skewe eformation (Fig. 4b), the compute F with magnitue F along irection shoul be further ecompose to a normal component F cn (i.e., normal to the originally uneforme surface of the elastic object) an a tangential component F ct. Depening on the magnitue of F ct, following the same analysis as presente in Sect. IV.B, we can etermine whether the hel rigi object gets stuck or performs compliant motion as the single region contact occurs an obtain the corresponing total contact force F c. Next, we can fin an equivalent point contact to the single region contact in that the contact force response to that point contact is the same as the total contact force response of the single region contact F c. With known F c an m (the mass of the rigi object, which can be consiere as concentrate on the equivalent contact point), the equivalent eformation isplacement D of the equivalent contact point can be obtaine from equation (3). The position of the equivalent contact point before eformation can be consiere as at the geometric center of the projection of the contact region on the originally uneforme surface of the elastic object along the eformation irection. See Fig. 5. Using such an equivalence of a point contact to the original single region contact simplifies the shape renering of the eforme elastic object (see Sect. V). (a) (b) (c) () Fig. 5. Examples of quivalent contact point : (a) before normal eformation, (b) after normal eformation, (c) before skewe eformation, () after eformation V. SHAPE DEFORMATION MODELING AND GRAPHIC RENDERING In Sections IV, the contact force response from a contact point or a single contact region of the elastic object to the rigi object is moele. Now we consier how to moel the shape eformation occurre on the elastic object ue to such a contact. A. Global Deformation Moeling an Renering Since global eformation is ue to the sprea of stresses/strains cause by the contact force, in the case of a single region contact, we use the equivalent point contact (which gives the same contact force effect see Sect. IV.C) to compute the eformation just as in the case of a single point contact, as escribe below. Recall that we maintain the exact parametric surface moel of the uneforme elastic object (Section II). Because the elastic object without eformation consists of smooth (flat or curve) faces, which may be boune by smooth (straightline or curve) eges an vertices, its eforme faces an eges shoul also be smooth except at bouning vertices an the contact (or equivalent contact) point to minimize elastic energy. Therefore, our strategy to capture the global eformation of an elastic object inclues following two steps:

5 1) Compute the amount of eformation at each curvature extremal point on its bounary surface, which is either a vertex, where the curvature is iscontinuous, a point with local minimum or maximum curvature, or an inflection point with zero curvature [15]. These extremal points are obtaine from the exact parametric surface moel of the elastic object. 2) Do a eformation interpolation between these extremal points an the contact (or equivalent contact) point to obtain smooth eformation of faces an eges of the elastic object. Both steps can be performe in real time as etaile in the following. 1) Deformation computation at curvature extremal points: To perform the first step, we nee to fin the elastic force at each curvature extremal point which causes the eformation an compute the amount of eformation accoringly in realtime. For this purpose, we introuce a novel Beam-Skeleton Moel to capture the unerlying physics of elasticity efficiently: once a contact is forme, the stress an strain relations between the contact (or equivalent contact) point an each curvature extremal point on the elastic object is moele by a beam whose central line connects them, where beam parameters are etermine by the physical properties as well as the exact parametric surface properties of the elastic object. The collection of such beams forms what we call a beam skeleton. Fig. 6 shows an example of a beam skeleton on an elastic ellipsoi object. Fig. 6. Elastic Object Hel Object Contact Point Beam skeleton (in soli lines) on an elastic ellipsoi object Now we escribe how eformation is compute for a beam of length l (which is etermine base on the uneforme shape of the elastic object) bent at one en with the other en fixe base on the Bernoulli-Euler bening beam theory [16]. Establish the beam coorinate system as O-xyz as shown in Fig. 7, where the origin is set at the center point of the fixe en of the beam, an the x axis is along the central line of the beam before it is bent an pointing to the other en of the beam. The y axis is following the bening force irection an the z axis is orthogonal to both the x an y axis following the right-han rule. A point p on the central line of the beam before it is bent has coorinates (x,0,0). Once the beam is presse at the en that is not fixe, the beam bens, an the new coorinates of p is (x,y,z ) satisfying x = x y = F y EI z ( 1 2 lx2 1 6 x3 ) z = 0 where E is the Young s moules an I z is moment of inertia with respect to the z axis. At the en of the beam where x = l, the relation between the external force F y normally applie to the beam en an the eformation istance y is F y = 3EI z l 3 y (4) we can relate F y to the stresses on the beam: σ x = F y I z (l x)y σ y = σ z = τ yz = 0 τ yx = 1 F y [ φ 1 2(1 + ν) I z y + νz2 (1 + ν)y 2 ] τ zx = 1 F y φ 1 2(1 + ν) I z z where σ s are stresses, τ s are shear stresses, I z is the moment of inertia, an φ 1 is the bening function of the eformable object epening on the shape of the cross section of the beam for ifferent shapes, φ 1 is ifferent. We call the above case where the external force is normal to the beam a simple bening case. O y Fig. 7. x Schematic of beam bening In general, the external force applie to a beam at one en is not necessarily normal to the beam central line. In such a case, base on the Saint-Venant principle [17], we can ecompose the force into a normal force an a tangential one an ecompose the problem of relating the external force to beam eformation into two simpler cases: one is the above simple bening case, an the other is a simple compression/expansion case, which we escribe below. For the simple compression/expansion of the beam cause by a tangential external force F x, assuming the area of the cross-section of the beam is S, we can relate F x to the tangential eformation (i.e., compression or expansion) x at the beam en x = l with the following equation: F x = ES x (5) l we can also relate F x to the stresses of the beam: σ x = F x /S σ y = σ z = τ xy = τ xz = τ yz = 0 The total stresses at a point of a beam are the vector sums of the stresses from the simple bening case an simple compression/expansion case. In our application, the force F applie to a contact or equivalent contact point from the rigi object to the elastic object can be viewe as applie to the common en of a beam skeleton consisting of n beams connecting the common en to each curvature extremal point of the elastic object. The force F can be obtaine as opposing the contact force response from the elastic object (as compute in Section IV) with the same magnitue. Establish a coorinate frame O i -x i y i z i for each F

6 beam i (i = 1,...,n) of the beam skeleton such that the origin O i is locate at the i-th curvature extremal point, which is at the other en of the beam i, x i axis is along the beam central line before it is eforme an pointing to the common en, y i is on the plane etermine by the x i axis an the force F, an z i axis is orthogonal to x i an y i axes following the right-han rule. Now we can view F applie to the same (contact) point of the common en of all the beams in the skeleton as the sum of the forces F i applie to each beam i at the same point so that the eformation occurre at each beam can then be compute separately. Notice that F oes not have a z i component in the beam i s frame. Let F xi an F yi be the x i an y i components of F i. From (4) an (5), we can obtain the following relations among all F i s an the relation between each F i an the eformation of beam i at the common en of the beam skeleton expresse in beam i s frame: F x1 :... : F xi :... : F xn = x 1 l 1 F y1 :... : F yi :... : F yn = y 1 l 3 1 :... : x i l i :... : y i l 3 i :... : x n l n (6) :... : y n l 3 n Aitionally, the sum of all F i s shoul equal to F: n F i = F (8) i=1 From the 2n equations (6), (7), an (8), F i can be solve for each beam i. With the above metho, we can compute the stresses at the fixe en of each beam (which is centere at a curvature extremal point of the elastic object). Now imagine the fixe en of each beam is no longer fixe, the effect of the stresses will make the corresponing curvature extremal point move to a new position. The position change can be consiere as the eformation at such a curvature extremal point, which can be compute from those stresses. Now we escribe how to compute strains from stresses an subsequently compute the eformation at each curvature extremal point. In general, the eformation isplacement of a point on the surface of an object can be expresse parametrically as {u,v,w}, satisfying u = u(x,y,z) v = v(x,y,z) w = w(x,y,z) The strain at point (u,v,w) can be represente as ǫ x x 0 0 ǫ y 0 y 0 ǫ ǫ = z 0 0 u = z v γ xy y x 0 γ w yz 0 z y γ zx z 0 x The relation between the strain an stress for a point on a linear elastic object can be represente as (7) (9) {ǫ} = D 1 {σ} (10) where D 1 is the inverse of the elastic coefficient matrix [16]. Given a curvature extremal point with coorinates {x, y, z} an its stresses, we can get its strains from (10). Next, its isplacement ue to eformation {u,v,w} can be solve from these strains using (9). 2) Global eformation renering: With the isplacements of all curvature extremal points of the elastic object ue to eformation etermine together with the isplacement of the contact (or equivalent contact) point, we obtain the eforme shape of the entire elastic object by an interpolation metho extening the Phong shaing metho [18]. Phong shaing is use for linear interpolation of vectors at vertices bouning a polygon across internal points of the polygon. In our case, however, the elastic object can have a general surface with curve features with or without eformation. Let P be the set of curvature extremal points plus the contact point of the surface of the elastic object before the current eformation, note that the elastic object can alreay be eforme before the current eformation to further change its shape. Such a surface can be partitione by curves connecting each pair of points in P into smooth curvature-monotonic surface patches, calle faces [15]. The vertices of each face are a subset of points in P. Now given the eformation isplacement vectors of these vertices for each face, we exten the Phong shaing to obtain a linear interpolation of the eformation isplacement vectors across the face as the following: Let θ i enote the angle between the irection of isplacement an the outwar normal irection of point i on the face, calle the isplacement angle of point i; the irections of isplacement vectors across the face are obtaine by linearly interpolating the isplacement angles of the vertices of the face, an the magnitues of isplacement vectors across the face are obtaine by linearly interpolate the isplacement vectors magnitues of the face s vertices. We obtain the eforme shape of the entire elastic object by performing the above shaing on all faces. Note that since we o graphical renering of an object base on its polygonal mesh approximation, the interpolate points of eformation shaing of a face can be simply the corresponing mesh points of the face. It shoul be emphasize that using our beam-skeleton moel to compute eformation fits the physics of elasticity well. First, accoring to physics, the stresses/strains are smoothly sprea over the elastic object surface an have extremal values on the curvature extremal points. Thus, if we can get the stresses/strains on all extremal points of the eformable object, we can get stresses/strains on any points between these extremal points by interpolations. Secon, the eformation isplacements at these extremal points with respect to the contact region can be thought of as the combination results of extension/compression, bening an twisting, all of which can be capture by the beam-bening moel. The global shape eformation of the elastic object obtaine from interpolating these isplacements satisfies that the closer a surface point to

7 the contact point or the equivalent contact point, the greater the eformation an curvature change is at this point. B. Local neighborhoo eformation moeling an renering For the cases of a single point contact, there is no nee to moify the result of global shape eformation escribe above in the neighborhoo R of the contact point. For the cases of a single region contact, however, the effect of the contact area on shape eformation in the neighborhoo R nees to be taken into account. Since the number of mesh points n in the contact region is proportional to the area of the contact region, we can efine a moification factor 1 w = 1+logn to capture the effect of the contact area on eformation: the larger the area, the shallower the eformation. Now, we use a moifie equivalent eformation isplacement wd (where D is etermine in Section IV.C), which is shallower than D, to conuct the same kin of interpolation as in global eformation renering within the neighborhoo R on the globally eforme shape to further moify the shape. The result is the combine effect of global shape eformation with local neighborhoo shape moification. Fig. 8 shows two examples of local neighborhoo shape eformation cause by a single region contact as moele by this metho. p c p c Fig. 8. Two examples of local neighborhoo shape eformation (p c inicates the moifie equivalent point, otte lines inicate the shapes before moification) VI. DEALING WITH MULTIPLE CONTACT REGIONS When there are multiple contact regions (at one time step), S i,i = 1,...,m, we establish a beam-skeleton moel with respect to each contact region in orer to compute both the contact force responses an the shape eformation of the elastic object. The beam-skeleton moel B i w.r.t. S i connects the (equivalent) contact point p i of S i to each extremal point of the elastic object as well as the (equivalent) contact point of every other contact region p j (j i) by beams. The contact force F Si from the elastic object to the rigi boy at contact region S i consists of not only the force from eformation at S i alone as escribe in Section IV but also the stress contributions from every other contact region S j, (j {1,...,m} {i}). The stress force contribution from S j can be compute from the beam-skeleton B j in a way similar to that escribe in Section V.A. The overall shape eformation of the elastic object can be compute by superimposing the shape eformation contribute by each contact region S i,i = 1,...,m, which can be compute base on each beam-skeleton moel B i in a way similar to that escribe in Section V. The only ifference is that here the (equivalent) contact points p j s (j {1,...,m} {i}) are fixe beam ens that o not move so that there is no nee to compute strains at those points. Fig. 9 shows an example with two contacts an their beam-skeletons on an elastic ellipsoi. Fig. 9. p 2 Elastic Object Beam skeletons for two contacts (in soli lines) on an elastic ellipsoi VII. IMPLEMENTATION AND TEST RESULTS We have implemente our approach an applie it to realtime haptic renering involving a virtual rigi boy an an elastic object via a PHANToM Premium 1.5/6DOF evice, which is connecte to a computer with ual Intel Xeon 2.4GHz Processors an 1GB RAM. Human operator can virtually hol the rigi object A by attaching it to the haptic evice an make arbitrary contact to the elastic object B (with its bottom center fixe, where a worl coorinate system is set) by guare motions an perform compliant motions on the elastic object. We use the following values for the parameters: m = 1kg, β 2 0ǫ = 0.4, ω 0 = 0.2, K = 10, β = 0.75, an parameters for rubber: ρ = 1100Kg/m 2, E = N/m 2, an ν = 0.5. Figures 10 to 12 show some test results, where the unit of force is Newton, the unit of length is mm, an the unit of time is ms. The worl coorinate system in all examples is built as the following: x-axis points right, y-axis points up, an z-axis points out of the paper plane an is orthogonal to x an y axes following the right-han rule. Fig. 10 shows a test example with a rigi mallet an an elastic ellipsoi. The human operator first move own the mallet to make a contact with the elastic ellipsoi, an move the mallet compliantly along the +x irection mostly an Force components (Newton) p Time (ms) Fig. 10. Force renering results when a mallet move compliantly along an ellipsoi fx fy fz Fx Fy Fz

8 Fig. 11. contact Moifie D (cm) Tail Contact 1.2 Hea Contact Force (Newton) Comparison of eformation in two ifferent cases of single region slightly towar the +z irection. In this case the contact normal generally has three components: F x, F y, an F z. Here uring the compliant motion, the value F x change from negative to positive, F y was always positive an reache the maximum value when the mallet was move to the upmost position of the ellipsoi, an F z only ha a small positive value since the movement was slightly on the half of the ellipsoi towar the +z irection. For the friction force f x, f y, an f z, f x always ha a negative value since the movement is towar +x irection, f y increase from negative to positive since the movement was first upwar then ownwar along the y axis, an f z was almost zero while the oscillation was from the ifficulty of moving the mallet in a straight motion. Fig. 11 shows an experiment to compare the ifferent effects of two cases with ifferent contact areas on local neighborhoo shape eformation. In case 1, a mallet s hea contacte an elastic cube, an in case 2, the mallet s tail contacte the elastic cube. The contact area in case 1 was larger than the contact area in case 2. We can see that when the same force was applie, the moifie equivalent eformation isplacement D of case 1 is smaller than that of case 2, which resulte in shallower eformation that fits the relate physics principle. Figure 12 shows example with two contact regions, where a rigi compass first touches a eformable heart with one sie pin an then both sie pins. We have use many test examples to confirm the generality of our presente metho of computing an renering contact force an shape eformation. In all of our experiments, moeling an computing haptic force took a constant an almost instant time, of approximately 30 µs, that is, the computation ha an upate rate of approximately 33 khz regarless of the objects geometry. This was negligible compare to the time neee for real-time contact etection (Section III) plus shape upating for the elastic object, which was in the orer of khz. In all test examples shown above, as can be seen, shape eformation of the elastic object looks quite realistic. VIII. CONCLUSIONS We have introuce a novel approach to moel an rener in real-time both the nonlinear contact force response an the shape eformation of a general elastic object cause by a rigi object contacting it an moving compliantly on it, taking into account friction. Our approach achieves both realtime efficiency an physical accuracy by taking avantage of nonlinear physics equations, elasticity principles, beam bening theory, an geometrical properties of general surfaces. The approach is implemente to confirm its effectiveness. An upate rate of over 1 khz is achieve for the entire renering process, incluing collision etection an renering both haptic force an graphic shape change. A future research goal is to further test the valiity of our approach. Fig. 12. Two contact regions between a rigi compass an a eformable heart: (left) one contact region (right) two contact regions ACKNOWLEDGMENT This work is partially supporte by the National Science Founation Grants EIA an IIS REFERENCES [1] S. Gibson an B. Mirtich, A survey of eformable moeling in computer graphics, TR-97-19, Mitsubishi Electric Research Lab, [2] A. Al-khalifah an D. Roberts, Survey of moelling approaches for meical simulators, in Proc. 5th Intl Conf. Disability, Virtual Reality an Assoc. Tech., (Oxfor, UK), pp , [3] C. Menoza an C. Laugier, Simulating cutting in surgery applications using haptics an finite element moels, in VR 03, pp , [4] F. Conti, O. Khatib, an C. 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