Haptic Simulation of a Tool In Contact With a Nonlinear Deformable Body

Transcription

1 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Haptic Simulation of a Tool In Contact With a Nonlinea Defomable Body Mohsen Mahvash 1 and Vincent Haywad 2 1 Real Contact Inc. Lone Cescent, Montéal, Québec, Canada H2X 2B1 2 Cente fo Intelligent Machines McGill Univesity, Montéal, Québec, Canada, H3A 2A7 {mahvash,haywad}@cim.mcgill.ca Abstact. This pape pesents a method to atificially e-ceate haptic feedback while moving and sliding an abitay vitual tool against a vitual defomable body with nonlinea elastic popeties. The computation of the esponse in such geneal cases is a task which does not yet admit computational solutions suitable fo ealtime implementation. To addess this, we descibe an appoach based on the bookkeeping of focedeflections cuves stoed at the nodes of a tiangulated body suface. Fo ealism, nomal and lateal defomations at each node ae epesented in a ange of deflection distances. The esponse eveywhee is synthesied via aea intepolation of esponse cuves stoed at the nodes of the mesh. The mathematical continuity of the synthetic esponse is the esult of both local coodinates intepolation and of esponse function intepolation, which pevious methods did not account fo. This guaantees the absence of haptic clicks and pops which ae unacceptable atifacts in high fidelity simulations. Sliding contacts ae also consideed. 1 Intoduction A haptic display is capable of e-ceating atificially the foces that simulate the inteaction of a tool with a defomable body. This capability can be used fo vaious applications such as in taining fo sugical tasks pefomed with a tool. The computational engine of a haptic display, namely the haptic engine, should be able to povide these foces in eal time. The constuction of a haptic engine that can simulate the continuous mechanics of the contact between a tool and a nonlinea defomable body in ealtime is a daunting task. Instead, we intoduce a pe-calculation appoach. It elies on the offline detemination of foce-deflection cuves at the nodes of a tiangulated body suface mesh fo a ange of deflection in a pe-pocessing step. Online, foces ae calculated at abitay locations via aea intepolation. The pe-calculation method appoach can be viewed as a systems appoach. Contact foces ae the outputs of a multi-input multi-output system. The inputs ae the position and the oientation of the tool tip, and the intenal vaiables epesent the position of the contact point. The behavio of the system is epesented by a constuction of piecewise polynomial cuves designed to appoximate

2 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Mohsen Mahvash and Vincent Haywad the behavio of an actual defomed body. The coefficients of these polynomials ae the system paametes. The systems fomulation allows us ensue two popeties fo the simulation which ae also popeties of the oiginal physical system, namely, continuity and passivity [15, 14]. Continuity is povided by the intepolation of the coodinates defined at the suface of vitual body and by the intepolation of the esponse functions. The identification of the system paametes is done offline. It is pefomed by fitting the outputs of the system to actual contact foces obtained though measuement o via accuate offline simulation. The paametes can be stoed in standad data file fomat since at any given time, only a small numbe of them is needed fo simulation. Section 2 eviews pevious wok. Section 3 descibes a mathematical famewok to calculate nomal and lateal foces. Section 4 explains how the pecalculated esponses can be constucted fom actual o accuate foces at a set of test points. Section 5 evaluates the intepolation appoach though a simple implementation. Section 6 concludes the pape. 2 Related Wok Vaious techniques wee poposed to make the haptic simulation of contact inteaction possible. We divide them into two goups. 2.1 Pe-calculation Methods Cotin et. al., James and Pai, and Bo-Nielsen intoduced the pe-calculation method [4, 11, 12, 16]. Linea models, obtained eithe fom the finite element method o fom the bounday element method, epesent the defomation of a body. A set of algebaic linea elations among the nodal quantities at the fee bounday ae then deived [12]. The defomation esponses given by unit foces (o tactions) applied to each node of the fee bounday ae calculated duing a pepocessing phase. The nodal foces in the contact egion ae calculated in advance fom a educed-ode linea system that diectly elates the nodal displacements to foces. A small linea system is deived fom the displacement esponses at untime. The defomation esponse of the bounday due to a displacement constaint imposed by the use is calculated as the supeposition of esponses of each nodal foce in a peset contact egion. Delingette et. al. as well as Astley and Haywad used pe-calculation techniques to compute the contact between a pat of a body (peipheal to the inteaction point) and the est of the body in a hybid body stuctue [7, 3]. Pe-calculation methods ae vey effective fo educing the computation time of defomation. Howeve, they ae applicable only to linea elastic bodies and small displacements. Due to discetiation, even fo linea elastic defomation, accuate computation of the foce deflection esponse equies a lage numbe of elements to epesent a body well. In the contact egion, this numbe can be abitaily lage, and hence can yield a pohibitively lage computational load.

3 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Simulation of Contact with a Nonlinea Defomable Body 3 In contast, the appoach intoduced in this pape is capable of handling nonlinea defomation (aising eithe fom lage defomation o fom nonlinea elasticity) due to massive pe-computation of the foces esponses in a pe-pocessing step. Moeove, the pe-computation buden can be educed by computing foces fo a mesh stuctue coase than the mesh stuctue used duing the simulation. 2.2 Time Integation Multi-Resolution Methods High ode dynamic defomation models epesent the defomation of the body. Explicit o semi-explicit time integation ae used to solve these models efficiently in ealtime. Multi-esolution calculations in space and time ae used to efine the calculation in egions of inteest. Astley and Haywad poposed multi-laye finite element mesh fo modeling defomation of a viscoelastic body [2]. Multi-ate integation could be used to compute the model. Zhang and Canny used the explicit time integation method fo haptic endeing of lage defomation [18]. D Aulignac et al. used implicit integation in a paticle-based method [5]. Wu et. al. used explicit integation and adaptive meshing fo simulating lage defomations of non-hookean mateials [17]. Debunne et. al. used adaptive multi-ate integation with multispatial esolution fo simulating tool contact with viscoelastic mateials. These appoaches essentially simplify eithe the finite element method o the contact poblem to make high ate computation of foces possible [6]. 3 System fo Tool Contact Simulation Figue 1 defines a haptic engine system fo a single tool. The inputs to the system ae x 1, x 2, x 3, the position components of the vitual tool tip (o the end effecto of the device) and θ 1, θ 2, θ 3 the oientation components of the tool tip. The intenal vaiables of the system ae the position components c 1, c 2, c 3 of a contact point c. Output foces and toques ae functions both of the inputs and of the intenal vaiables. This epesentation of the haptic engine system assumes a quasi-static contact. It also assumes that the contact aea between the tool and the body always contains the tool tip. x 1 x 2 x 3 q 1 q 2 q 3 c 1 c 2 c 3 f 1 f 2 f 3 t 1 t 2 t 3 Fig. 1. Haptic engine. In the following, we descibe the calculation of output foces fo a haptic engine. A simila appoach can be used fo the calculation of output toques. We ignoe the dependency of contact foces on tool oientation. It could be included following exactly the same appoach.

4 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Mohsen Mahvash and Vincent Haywad 3.1 Nomal Contact The haptic engine eceives x(t), the position of tip of a vitual tool. The contact point c epesents the position of the initial contact of the tool on the suface of the defomable body at est. The deflection vecto at point c is δ = x c, see Figue 2 and caption. a b c = x(t) c d x(t) Fig. 2. (a) A tool initially contacts a body at point c. (b) With penetation, a contact suface is fomed (shown as a cicle). The known tool tip position x gives an appoximation fo the position of this suface. The deflection δ at point c, a point of initial contact on the suface of an undefomed object (in black outline), is found fom the diffeence between the position of the contact suface and c. When the object defoms (in black outline), a foce esponse is poduced. Nodes distibuted on the suface of the body can be oganied in a tiangulation used to epesent the geomety of the body at est as a set of tiangula elements. With the foce esponse pe-calculation appoach, the need to detemine the nodal displacements online is eliminated. Response of a defomable body. The foce esponse to contact at each node l of element m is epesented in local coodinates. Each node is labeled x l,m. The unit vectos u l,m and u l,m with oigin at x l,m ae such that δ l,m = δ l,m u l,m + u l,m δ l,m, as shown in Figue 3a. The quantities f l,m (.) and f l,m (.) ae the components of the foce esponses as in Figue 3b. The esponses can be deived fom actual foce esponses (pe-calculated o pe-measued with suitable appoximations [14]) using piecewise polynomial intepolation. Given x l,m and x, the foce esponse at node l of element m is: f l,m (δ) = f l,m(δ l,m) u l,m + f l,m(δ l,m) u l,m. (1) a d l,m u l,m d x c = x d l,m l,m u l,m b f l,m (d l,m ) d l,m f l,m (d l,m ) d l,m Fig. 3. (a) Deflection in local coodinates. (b) Response vecto components.

5 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Simulation of Contact with a Nonlinea Defomable Body 5 Intepolated esponse. When c is inside an element m, coodinates ae defined at point c with a unit vecto u c found by intepolation of unit vectos at each node. The unit vecto u c is such that δ = δc u c + δcu c as in Figue 4. Call n l,m (.) the intepolation function fo node l of element m: u c = n l,m (c) u l,m. (2) l=1,2,3 u 2,m u c u 1,m d c c d c d u c u 3,m Fig. 4. Local coodinates at c. The nomal vecto u c points inside the body. We use a common intepolation scheme used in computational methods fo continuous mechanics based on natual coodinates. Within one element of aea A m, these coodinates ae given by the aeas A l,m of the tiangles defined by the contact point and the thee vetices as in Figue 5a, n l,m (c) = A l,m(c) A m. (3) Note that the vecto u c is nomally distinct fom the vecto nomal to the suface of the element. A key popety of the intepolation appoach is to ensue continuity of coodinate change ove the suface of the body. Refeing to Figue 5b, the focedeflection esponse at c is obtained fom: f c (δ) = f c (δ c )u c + f c (δ c)u c, (4) whee fc (.) and fc (.) ae also intepolated fom thee pe-calculated focedeflection esponses: fc (.) = n l,m (c) fl,m(.), (5) f c (.) = l=1,2,3 l=1,2,3 n l,m (c) f l,m(.). The haptic engine outputs ae now defined fo any initial contact point and tool tip position.

6 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Mohsen Mahvash and Vincent Haywad a 2 1 A 2,m A 3,m A 1,m 3 b f 2,m f 2,m m d 2,m d 2,m d 2,m u 2,m u c c u 3,m u 1,m Fig. 5. (a) A point inside a tiangle defines thee aeas labeled as indicated. (b) Fo each tiangula element m, the vetices ae associated to u l,m, l = 1, 2, 3 and to two esponses f l,m and f l,m. Similaly, vaiations in the limiting coefficient of fiction as a function of penetation can be encoded ove the suface of the body. 3.2 Sliding Contact Sliding behavio is epesented in tems of pe-sliding displacements [10]. Duing sliding contact, c moves ove the body bounday. Let δ s be the limiting sliding deflection such that f c (δ s) = µ(δ c )f c (δ c ), whee µ(.) is the limiting fiction coefficient given as function of nomal penetation. The pe-sliding distance is not to be confused with the pe-sliding obseved with had mateials in contact, which is measued in µm [1]. Fo soft mateials, it can be lage and depends mostly on the mateial popeties and on its suppot. Conside, fo example, the case of palpation. The skin can defom ove lage distances befoe slip occus. Applying Coulomb s law is equivalent to assuming that µ is invaiant with the contact suface, i.e., is independent fom penetation. In the pe-sliding model, sliding occus when δ > δ s [10]. Duing sliding, c moves ove the body suface in such manne that δ δ s at all times. The movement is in the diection of the pojection of u c ove the element suface and its magnitude is δ c δ s, see Figue 6. u c c Dc = d c -d s u c Fig. 6. Vecto u c is pojected onto the element suface to define the diection of the movement of c. If δ c eaches δ s, c slides by δ c δ s.

7 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Simulation of Contact with a Nonlinea Defomable Body 7 4 Constuction of the Pe-calculated Responses The pe-calculated foce-deflections f l,m (δ ) and f l,m (δ ), as well as the fiction popeties descibed by µ(δ ), can be intepolated by piecewise linea o polynomial intepolatos. The paametes of the intepolatos ae calculated such that the outputs of the haptic system fit eithe the actual foces obtained by expeimental measuement o accuate foces geneated by an offline simulato [14, 9]. The location and numbe of testing points can diffe fom those of the suface nodes used to epesent a vitual body. Fo example, Figue 7 illustates how a mesh can be efined by obtaining the foce esponse at a new node fom pecalculated foces at thee test points, using an intepolation appoach simila to the one of the last section. Moe geneally, the numbe of the test points fo a body can be much smalle than the numbe of vitual body suface nodes used fo a simulation. This is useful when it is known that the change of foce esponse ove the suface of a body is smoothe than, fo example, the change in visual aspect o in the geomety of the body suface. Test Data New node Fig. 7. Foce Deflection esponses fo a new node of a mesh of a body suface is obtained by foce-deflection esponses of thee test points using aea intepolation. 5 Implementation A simple haptic engine system was implemented to evaluate the pefomance of the intepolation appoach. The vitual envionment compised a tool inteacting with a defomable cylindical vitual body made of about 500 patches. The local esponses at a node l of element m wee: { 0 δ fl,m(δ 0 )= 100 kl,m δ2 0 < δ (6) f l,m(δ ) = k l,m δ (7) The vectos u l,m at each mode wee in the adial diection.

8 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Mohsen Mahvash and Vincent Haywad Fig. 8. Gaphical use inteface and device used in the tool contact simulation. The test showed that coodinates intepolation, togethe with foce-deflection intepolation povided continuity fo the esponses. Convesely, eliminating one of the two intepolations caused a vaiety of atifacts, including unwanted textues o limit cycles esulting fom lack of passivity duing sliding motions. These could be attibuted to discontinuities in deflection o in the poduced foce at the mesh edges. These unwanted atifacts wee called haptic clicks and pops by analogy to the defects in sound poduction. Figue 8 shows the use inteface used in the test. The simulation pogam consisted of two independent eal-time theads unning unde RTLinux-3. One thead povided fo endeing the foces and the othe fo finding the active element. The foces wee geneated by a PenCat/Po TM haptic device (Immesion Canada Inc.) which is diect dive and hence povides good fidelity because of nea absence of mechanical damping. 6 Conclusion This pape descibed a pe-calculation solution fo haptic simulation of contact between a tool and a nonlinea defomable body. The appoach elies on pe-calculated foces esponses known at nodes of the tiangulated body suface mesh. It calculates foces at abitay locations though intepolation of both foce esponses and coodinates. Accoding to the needs and the application, piecewise linea o polynomial intepolatos can be used to egiste nomal and lateal foces at each node at each diection fo a ange of deflection distances. The pe-calculation esponses can be deived fom actual foce esponses esulted fom measuement o accuate offline simulations. The measuements o simulations can be done only at a set of testing point located ove the suface of the body and the esults ae extended to many nodes of the body suface by using aea intepolation.

9 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Simulation of Contact with a Nonlinea Defomable Body 9 This appoach could be genealied to cases that include damage and plasticity by adding additional intenal vaiables. The case of cutting, a special case of damage, was teated in [13]. The systems appoach also allows us to study stability of haptic inteactions in a geneal way [15]. Acknowledgement This eseach was funded by the poject Reality-based Modeling and Simulation of Physical Systems in Vitual Envionments of iis, the Institute fo Robotics and Intelligent Systems (Canada s Netwok of Centes of Excellence). Additional funding is povided by nsec, the Natual Sciences and Engineeing Council of Canada, in the fom of an opeating gant fo the second autho. Refeences 1. Amstong-Hélouvy, B., Dupont, P., and Canudas De Wit., C A Suvey of Models, Analysis Tools and Compensation Methods fo the Contol of Machines with Fiction. Automatica, 30(7): Astley, O. R., Haywad, V Multiate Haptic Simulation Achieved by Coupling Finite Element Meshes Though Noton Equivalents. Poc. IEEE Intenational Confeence on Robotics and Automation, pp Astley, O., Haywad V Design Constaints fo Haptic Sugey Simulation. Poc. IEEE Intenational Confeence on Robotics and Automation, pp Cotin, S., Delingette, H., Ayache, N Real-time Elastic Defomations of Soft Tissues fo Sugey Simulation. IEEE Tansactions on Visualiation and Compute Gaphics, Vol. 5:1, pp D Aulignac, D., Balaniuk, R., Laugie, C A Haptic Inteface fo a Vitual Exam of the Human Thigh. Poc. IEEE Intenational Confeence on Robotics and Automation, pp Debunne, G., Desbun, M., Cani, M., Ba, A Dynamic Real-Time Defomations Using Space and Time Adaptive Sampling. Compute Gaphics and Inteactive Techniques, SIGGRAPH 2001, ACM Pess, pp Delingette, H., Cotin, S., Ayache, N A Hybid Elastic Model Allowing Realtime Cutting, Defomations and Foce-feedback fo Sugey Taining and Simulation. Compute Animation Poceedings, pp Fank, A. O., Twombly, A. I., Bath, T. J., Smith, J. D Finite Element Methods fo Real-Time Haptic Feedback of Soft-Tissue Models in Vitual Reality Simulatos. Poc. of the Vitual Reality Confeence, pp Geenish, S., Haywad, V., Chial, V., Okamua, A., Steffen, T Measuement, Analysis and Display of Haptic Signals Duing Sugical Cutting. Pesence: Teleopeatos and Vitual Envionments, MIT Pess. Vol. 6(11). pp Haywad, V., Amstong, B A New Computational Model of Fiction Applied to Haptic Rendeing. In Expeimental Robotics VI, P. I. Coke and J. Tevelyan (Eds.), Lectue Notes in Contol and Infomation Sciences, Vol. 250, Spinge-Velag, pp James, D. L., Pai, D. K AtDefo, Accuate Real Time Defomable Objects. SIGGRAPH 99 Confeence Poceedings, pp

10 In IS4TM: Intenational Symposium on Sugey Simulation and Soft Tissue Modelling Lectue Notes in Compute Science (LNCS 2673), N. Ayache and H. Delingette (Eds), Spinge Velag, New Yok pp Mohsen Mahvash and Vincent Haywad 12. James, D. L., Pai D. K A Unified Teatment of Elastostatic and Rigid Contact Simulation fo Real Time Haptics. Haptics-e, the Electonic Jounal of Haptics Reseach, Vol. 2, No Mahvash, M., Haywad, V Haptic Rendeing of Cutting, A Factue Mechanics Appoach. Haptics-e, the Electonic Jounal of Haptics Reseach, Vol. 2, No Mahvash, M., Haywad, V., Lloyd, J. E Haptic Rendeing of Tool Contact. Poc. Euohaptics pp Mahvash, M., Haywad, Passivity-Based High-Fidelity Haptic Rendeing of Contact Poc. IEEE Intenational Confeence on Robotics and Automation, in pint. 16. Bo-Nielsen, M Finite Element Modeling in Sugey Simulation. Poceedings of the IEEE, 86:3, pp Wu, X., Downes, M. S., Goktekin, T., Tendick, F Adaptive Nonlinea Finite Elements fo Defomable Body Simulation Using Dynamic Pogessive Meshes. Poc. Euogaphics 2001, Vol. 20, No. 3, pp Zhuang, Y., Canny, J Haptic Inteaction with Global Defomations. Poc. IEEE Intenational Confeence on Robotics and Automation, pp