Cutting on Triangle Mesh Local Model based haptic display for dental preparation surgery simulation

Cutting on Triangle Mesh:Local Model-Based Haptic Display forDental Preparation Surgery SimulationDaniel Wang,Yuru Zhang,Member,IEEE,Yuhui Wang,Yuan-Shin Lee,Peijun Lu,and Yong WangAbstract—A new method to realize stable and realistic cutting simulation using an impedance display haptic device andmicrocomputer is presented in this paper.Material removal or cutting simulation is a critical task in dental preparation surgery simulation.In this paper,a piecewise contact force model is proposed to approximately describe the cutting process.Challenging issues of minimizing the difference between the cutting simulation and haptic contact simulation are analyzed.The proposed contact-based simulation method is developed for a one-dimensional cutting task and can be expanded to three-dimensional cases.Local model-based multirate simulation cutting architecture is proposed and force control of the haptic device is decoupled from the cutting simulation loop,which can both ensure high fidelity of dynamical simulation as well as maintain stability of the haptic device.The cutting operation is realized using spherical and cylindrical shaped tools.An experiment based on the Phantom desktop proves that fidelity in one-dimensional cutting can be realized and stability in three-dimensional cutting can be ensured using the force-filtering method.Index Terms—Dental surgery simulation,haptic display,cutting simulation,impedance display,piecewise contact force model,local model,multirate simulation,stability.æ1I NTRODUCTION1.1BackgroundI N dental surgery,typical operations include cutting the tooth using a high-speed rotating tool,which is essential to performing cavity preparation and other procedures. During these operations,haptic sensation is important for the surgeon to implement the surgery successfully.Dental students currently use artificial teeth and jaws or visual-only computer simulations to practice the operation.Plastic models cannot provide the level of detail and material properties of real-life teeth and procedures[1].Sensorimo-tor skill in nature is lost in classical visual-only computer simulations.Therefore,haptically enhanced simulation is proposed to provide the sensorimotor involvement needed for dental training.Some pioneering work has been done in the haptic-based dental training field.Ranta et al.introduced a concept design of a Phantom-based dental training system to practice cavity preparation[1].Thomas et al.developed a training system with the Impulse2000,enabling the operator to practice the detection of carious lesions[2].Their system focuses on probing the difference between different tissues on a tooth,while cutting simulation is not considered.Compared with traditional haptic contact simulation, special constraints must be met for cutting-related haptic surgery simulation[3],[4].In cutting simulation,tissues will be removed according to the interaction between the moving tool and tissues.This feature imposes a challenge in the computation of interaction force.Considering the goal of dental cutting surgery simula-tion and the characteristics of the dynamical cutting process,challenges in cutting simulation can be summar-ized as follows:plex interaction dynamics:Interaction forceduring cutting is influenced by multiple variables,such as the rotating dynamics of the tool,the heateffect from high speed machining,and the fracturemechanism of the tooth,etc.2.Virtual environments(VE)with complex properties:The geometry of a tooth surface is heavily curvedand different tissues exist inside the tooth.Mappingfrom the motion signal of the haptic device to thevirtual force signal is nonlinear.3.Real-time evolving VE:The shape of the tooth isevolving because of material removal from the tooth’ssurface.How to describe the relationship between theremoved material and the interaction force whilemaintaining stability is another challenge.4.Effect of a cutting tool’s shape on the cutting forcecomputation:Multipoint collision detection must beconsidered and material removed is decided byshape of the tool’s surface.The updating rate of the. D.Wang,Y.Zhang,and Y.Wang are with the Robotics Institute,BeihangUniversity,No.37Xueyuan Road,Haidian District,Beijing,PR China.E-mail:hapticwang@,yuru@,wangyuhui@..Y.-S.Lee is with the Department of Industrial Engineering,North CarolinaState University,Raleigh,NC27695.E-mail:yslee@..P.Lu and Y.Wang are with the School of Stomatology,Peking University,Beijing,PR China.E-mail:kqcadc@.Manuscript received5Dec.2004;revised29Apr.2005;accepted12May2005;published online8Sept.2005.For information on obtaining reprints of this article,please send e-mail to:tvcg@,and reference IEEECS Log Number TVCGSI-0161-1204.1077-2626/05/$20.00ß2005IEEE Published by the IEEE Computer Societysimulation may be decreased because of the largecomputation and,also,stability will be decreasedbecause of the effect.5.Unpredicted movement of a cutting tool:Thetrajectory of the cutting tool is generated by real-time motion from the operator,which can bearbitrary.The effect of the various feeding displace-ment and feeding velocity in the three-dimensionalworkspace on interaction force and material removalneed to be reflected precisely.Considering the above challenges,the real challenge lies in how to achieve balance among fidelity,stability,and computation efficiency in the dental cutting simulation. Several key tasks,such as object modeling,collision detection,haptic rendering,model update,etc.,must be considered simultaneously.This paper focuses on addres-sing these issues.1.2Literature ReviewAgus et al.introduced development of haptic-enabled bone cutting simulation systems[5],[6].In their system,the volume model is used for the bone and the tool is simplified to be a sphere and a realistic cutting simulation result is pared to polygon model-based methods,the volume models impose stricter requirements for the computer’s memory and computational efficiency.A dual processor graphical display computer is usually needed to ensure real-time direct volume rendering.Petersik et al.[7] model a drilling instrument as a point cloud and use a modified version of the Voxmap-Pointshell algorithm[8]to sample the surface of the drill and generate appropriate forces at each sampled point.Morris et al.also work on a haptic/visual bone dissection simulation involving removal of material[9].Hybrid data representation that allows smooth surfaces is used for graphic rendering,while volumetric data is used for haptic feedback.Mendoza et al.discuss three methods for deformation or material removal computation:mass-spring meshes,FEA, and LEA methods[10].The displacement of the tip of the haptic device is used directly as the input variable for the mesh,which doesn’t consider the finite impedance property of the haptic device.The tool’s shape is usually simplified into a point or a line in cutting simulation[10],[11],[12].Cutting on the clay using different shaped tools has been supported by the Freeform1system[13].Zhu and Lee proposed a haptic simulation algorithm for NC cutting and machining planning[14],[15].Collision detection is carried out between the trajectory of the point or the line and the object being cut,which is not precise enough for simulating a collision between the tool’s boundary and the tooth.These systems usually adopt the voxel-based or the dexel-based modeling methods,which impose a critical requirement for the hardware configuration and simulation algorithms.The virtual coupling and time domain passivity methods have been used to analyze the stability of the haptic system [16],[17],[18],[19],[20].Implementation of these methods in a complex VE simulation such as dental cutting is difficult.Barbagli et al.gave the local model stability issues in deformed object manipulation[21],however,only one dimensional experiment is given for the algorithm imple-mentation.The stability of the rendering algorithm on the triangle mesh cutting simulation has not been reported systematically.1.3Organization of the PaperIn this paper,triangular mesh is used for modeling the tooth and the cutting tool is simplified to be an analytical shaped object.A force model based on the penetration distance between the tool and the tooth is formulated.In this paper,a vertex deformation method is developed to realize material removal.The remainder of this paper is organized as follows: Section2proposes a piecewise contact force model for simulating triangle mesh cutting.Section3discusses the one-dimensional cutting simulation method.Section4 expands the method into the three-dimensional cutting simulation and incorporates a local model to achieve a trade-off among fidelity,stability,and computational efficiency.Section5gives implementation and some practical experiment results,followed by the concluding remarks and some future research work in Section6.2A NALYSIS OF I NTERACTION S TATUS DURINGC UTTING2.1Piecewise Contact ModelIn this paper,a piecewise contact model based on interaction status transition is proposed to describe the dynamic cutting process under the operator’s real-time cutting operation and to realize fast computation of the interaction force and material removal.Under this idea,the cutting process can be described as a series of transitions among different interaction statuses.During the cutting process in a simulation,the relative motion of a virtual tool against a tooth is determined by the real-time motion of the haptic device.Three interaction statuses are defined:1)separation status,2)contact status, and3)cutting status.Separation status refers to the status when the tool does not make contact with the tooth in the geometry space.The contact status refers to the status when the tool makes contact with the tooth where no material is removed from the tooth and the cutting status is the status when the tool makes contact with the tooth and material is removed from the tooth.The cutting process is described as the transition among the three different statuses and the transition map among the three statuses is shown in Fig.1,where~F n is the contact force and F Limit is a predefined force threshold representing the resistance force limit to remove material from the tooth.From clinical dental operation,it is found that the cutting force against a tooth is mainly affected by the following parameters:the volume of removed material,the tool’s feeding velocity,the contact status between the tool and the tooth,and the friction force.Furthermore,the tooth’s surface is updated during real-time cutting.Therefore,the mapping from the device’s motion to the virtual force is a complex nonlinear mapping.In control theory,a nonlinear system can be transformed into a linear system at a local working point.Similarly,the cutting process can be described as a series of local contactprocesses.At any two consecutive simulation time periods k À1and k ,if (1)is satisfied,the cutting force can be computed using the piecewise contact force model.j @F e =@X 0j <X h ðk Þ¼X h ðk À1Þ;X 0ðk Þ¼X 0ðk À1Þ;&ð1Þwhere F e is the virtual force,X h k À1ðÞ;X h k ðÞis the position of the haptic device’s tip at two consecutive simulation time periods,and X 0k À1ðÞ;X 0k ðÞis the tooth’s boundary at two consecutive simulation time periods. is a finite,user-defined small value.Equation (1)means that the change of the interaction force resulting from material removal is small enough while the haptic device is held in one static position.In another sense,the critical thing that ensures the validity of the piecewise contact model is to maintain a small material removal volume within each cutting simulation cycle.The validity of (1)will be verified by experiment in Section 5.3.2.2Digital Model of Tooth and ToolThere are two objects in the virtual environment,the tooth and the dental tool.The tooth is supposed to be static in the world coordinate system.The tool moves along with the movement of the Phantom.The tooth is modeled using triangle mesh.The original data is multilayered points sampled via a laser scanning data acquisition device,which defines the boundary shape of the tooth.A minimal diagonal distance method is used to reconstruct the points’data into a triangle mesh [22].Triangles on the mesh are assigned with a predefined tissue property,which will be used in the force computation and graphical display.In order to display the force difference on different tissues,such as enamel,dentin,pulp,and cavity,several additional triangle meshes can be defined to describe the boundary between different tissues.These meshes separate the volume inside the original tooth shape into several subvolumes.At the beginning of the cutting simulation,all triangles on the mesh belong to the enamel tissue type.During the cutting process,the triangle mesh of the tooth surface model is modified.For each newly produced triangle,a tissue type detection algorithm is used to decide the physical property of the triangle.The model of the probe is constructed first,using UniGraphics,then it is exported as a VRML ingGHOST SDK,the VRML model is transformed into a triangle mesh.The cutting tool is described using an analytical model.The geometric center of the model corresponds to the position of the Phantom tip.2.3Definition of Different Tools in Dental Cutting Haptic SimulationIn a typical force rendering method,the dynamic object is simplified as a point [23].During the interaction between a tool and a tooth,a collision between a tooth and the tool’s shape needs to be detected to improve force fidelity.Three tools,the Physical Tool ,the Virtual Tool ,and the Cutting Tool ,are defined to enable force computation and tooth material removal.Motion scaling between the physical tool and the virtual tool is utilized to prevent the virtual tool from being totally immersed into the tooth triangle mesh.The relation between the three tools is dictated in Fig.2,where the tool is sphere-shaped.The center of the PhysicalTool ,~Xp ,corresponds to the position of the Phantom tip position ~Xh .The center of the Virtual Tool ,~X V ,is computed using a motion-scaling method,which will be used to do collision detection with the tooth’s triangle mesh.The centerof the Cutting Tool ,~XC ,is consistent with ~X V ,whereas the size of the Cutting Tool is smaller than that of the Virtual Tool .This tool is used to compute the cutting section on the tooth’s surface.The computation flowchart during the cutting process is dictated as in Fig. 3.Because the maximum stiffness displayed by the Phantom desktop is limited,the Physical Tool and the Virtual Tool will interpenetrate the tooth to get a continuous changing virtual force,whereas the Cutting Tool should be stopped under the physical constraint of the tooth’s surface,that is,the movement of the Cutting ToolWANG ETAL.:CUTTING ON TRIANGLE MESH:LOCAL MODEL-BASED HAPTIC DISPLAY FORDENTAL PREPARATION SURGERY (673)Fig.1.Interaction status transition during the cutting process.Fig.2.Relation between the three tools.putation flowchart of the cutting process.represents the feeding movement of the cutting tool in the physical world.Motion scaling is used to compute the position of the Virtual Tool:_~XV¼_~X p=K S;ð2Þ~X V kðÞ¼~X V kÀ1ðÞþ_~X VÁT;ð3Þwhere K S is the motion scaling variable,~X V kðÞand~X VkÀ1ðÞare the positions of the Virtual Tool at the currentand previous time step,and T is the sampling period of the haptic loop.In Fig.3,the task of collision detection is to find those vertices on the tooth’s triangle mesh that penetrate into the Virtual Tool’s sphere.In order to realize a stable haptic display,the computation rate of the collision detection should be greater than1KHz.In this paper,a typical space partition method is used to accelerate the collision detection process between the Virtual Tool and the tooth triangle mesh[24].3O NE-D IMENSIONAL C UT AGAINST V IRTUAL W ALL 3.1Virtual Force ComputationFor the virtual wall illustrated in Fig.4,X0represents the boundary of the wall,~n is the normal vector of the wall,K et is the virtual stiffness,and~F e is the virtual force.The direction of~n is the opposite direction of axis X.The three-dimensional motion signal of the haptic device is measured and only a one-dimensional signal,for instance,X h,is used in collision detection and material removal computation.Based on the interaction status transition model,the cutting process can be regarded as series of contacting processes.In the literature,a unilateral spring force model is utilized to compute the virtual interaction force[25]F e kðÞ¼ÀK et X h kðÞÀX0kÀ1ðÞ½ X h kðÞ!X0kÀ1ðÞF e kðÞ¼0X h kðÞ<X0kÀ1ðÞ; &ð4Þwhere F e kðÞis the virtual contact force in the current simulation loop,K et is the virtual contact stiffness of the wall,X h kðÞis the position of the haptic device in the current simulation loop,X0kÀ1ðÞis the position of the virtual wall’s boundary in the previous simulation loop.3.2Model for the Material RemovalIn one-dimensional cutting,the shape of the tool is simplified as a point and the material on the virtual wall is removed along the inverse normal of the wall.When the interaction force grows bigger than a threshold,the status between a tool and a virtual wall evolves from contacting to cutting and the boundary of the virtual wall should be updated as follows:X0kðÞ¼X0kÀ1ðÞþÁX;ð5ÞÁX¼ÀF e kðÞ=K me F e kðÞ!FÃeÁX¼0F e kðÞ<FÃe;&ð6Þwhere X0kðÞ,X0kÀ1ðÞis the coordinates of the virtual wall’s boundary at the current and previous time steps,ÁX is the depth of removed material from the surface of the virtual wall,K me is the removing coefficient of the material,FÃeis the threshold of the force that is needed to remove material from the tooth.The position of the tool in the graphical scene is computed using the following equation:X G¼X h X h>X0X G¼X0X h X0:&ð7ÞTo ensure a synchronous display between the graphical and the force display,the position of the tool in the graphical scene needs to be updated according to the interaction status.3.3Stability AnalysisInside the contact status for one-dimensional interaction, Colgate et al.give a sufficient condition to maintain system stability[25]b>KT2þB j j;ð8Þwhere b is the inherent damping of haptic device,K;B is the stiffness and damping of the virtual environment,and T is the sampling period.For the virtual wall with pure stiffness,the upper limit of virtual stiffness isK<2bT:ð9ÞHowever,this criterion is only suitable for contact simulation.In one-dimensional cutting,the boundary of the virtual wall is changing.Therefore,new stability criteria need to be derived.Based on the interaction status transition model,the cutting process is equivalent to a series of several contact statuses.Therefore,stability in the cutting process can be transferred into two components:1)Stability must be ensured inside each contacting phase and2)stability must be ensured at the switching time between two contacting statuses.For the first requirement,according to(4)and(9),the following criteria can be derivedK et<KÃe;ð10Þ674IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS,VOL.11,NO.6,NOVEMBER/DECEMBER2005 Fig.4.Interaction with the virtual wall.where K Ãe is the upper limit of virtual stiffness in a specified one-dimensional axis,which can be measured via experiment.For the second requirement,suppose that:1)At time k À2,the haptic device locates inside the constraint space and the material removal does not take place and 2)at time k À1,the material removal takes place and the boundary of the virtual wall is moving.That is,X 0k À2ðÞ¼X 0k Ài ðÞX 0k À1ðÞ¼X 0k À2ðÞ&i ¼3;ÁÁÁ;k:ð11ÞAccording to (1),the virtual force at time k needs to satisfy the following criteria:F e k ðÞÀF e k À1ðÞ< :ð12ÞCombined with (4),thenF e k ðÞÀF e k À1ðÞ¼ÀK et X h k ðÞÀX h k À1ðÞ½ þK et X 0k À1ðÞÀX 0k À2ðÞ½ ¼ÀK et X h k ðÞÀX h k À1ðÞ½ ÀK etF e k À2ðÞK me:ð13ÞCombined with (12)and (13),it can be derived thatK me >ÀK et F e k À2ðÞþK et X h k ðÞÀX h k À1ðÞ½ f g:ð14ÞConsider the biggest influence that the change of the virtual wall’s boundary has on the change of the virtual force,that is,X h k ðÞÀX h k À1ðÞ¼0:ð15ÞThe lower boundary of K me is approaching the max-imum value.The second stability criteria for the one-dimensional cutting system can be derived asK me >ÀK et F e k À2ðÞ:ð16ÞTherefore,the stability criteria for the one-dimensional cutting system can be derived asK et <K ÃeK me >ÀK et F e k À2ðÞ :&ð17Þ4T HREE -D IMENSIONAL C UT S IMULATION B ASED ON THE D YNAMIC L OCAL M ODEL4.1Contradiction between Fidelity and Efficiency In order to realize a realistic simulation,the first challenge is to meet the demand for high computation efficiency.Realistic simulation may require large computation,includ-ing collision detection,physical-based interaction force computation,collision response,and update of the virtual object’s model.According to the requirement of a 1KHz force update rate in the haptic display [25],it is difficult to finish all the above computation in the specified simulation loop.Therefore,the contradiction between fidelity and computation efficiency needed to be solved.Barbagli et al.propose a local model to compute the deformation of soft tissues,which is effective in solving the contradiction [21].The finite stiffness of the haptic device is another fundamental constraint for selecting the simulation algo-rithm.In the literature,the spring-damping model is commonly used to realize the transform from the motion signal of the haptic device to the virtual force signal.With regard to a haptic device with the impedance display principle,such as Phantom,the stiffness that can be simulated stably is within a range.Vibration or noise will occur if the stiffness range is violated.Within the stiffness range,the position of the haptic device will penetrate into the surface of the tooth’s boundary when simulating a big interaction force.In the graphical display scene,the virtual tool should be constrained on the surface of the tooth.Therefore,virtual proxy or “God-object”[21],[23]must be computed in real time to get a consistent display between the graphical and the haptic loops.Furthermore,it is necessary to compute the cutting section surface and update the tooth model in real time.With regard to the above requirements,the local model method is utilized in the material removal simulation in this paper.4.2Principle of 3D Local Model-Based Cutting SimulationDuring the cutting simulation,the tip of the Phantom desktop is driven under the operator’s movement and collision detection is implemented between the virtual tool and the triangle mesh of the tooth based on the space partition bounding box method.In order to reflect the influence of the tooth’s shape on the interaction force,the tangent plane at the contact point between the virtual tool and the tooth is constructed as the local model.Under a different interaction status,the local model should reflect the tooth’s shape and stiffness characteristics on the contact point.As Fig.5a illustrates,the virtual tool locates in free space and the virtual force is zero and the local model doesn’t exist.In Fig.5b,the tool begins contacting the tooth and the local model is the tangent plane on the contact point.In Fig.5c,the position of the local model remains unchanged when the virtual tool moves forward and penetrates into the tooth.In Fig.5d,the position of the local model changes when the virtual tool slides along the tooth’s surface and the contacting status is maintained.In Fig.5e,the position of the local model changes when material is removed from the tooth’s surface.In Fig.5b,the given local model reflects the influence of the tooth’s shape and stiffness property.However,the virtual force cannot be computed directly using this model.BecauseWANGET AL.:CUTTING ON TRIANGLE MESH:LOCAL MODEL-BASED HAPTIC DISPLAY FOR DENTAL PREPARATION SURGERY...675Fig.5.Evolution of the local model during the cutting process.the tool is sphere-shaped instead of point-shaped,the local model must be translated to reflect the collision between the tool’s boundary and tooth’s surface.Illustrated by Fig.6a,the local model should be translated from position 1to position 2corresponding to the case in Fig.5b.Illustrated by Fig.6b,the local model should be translated from position 1to position 2corresponding to the case in Fig.5c.The architecture of the local model-based multirate simulation system is given in ing the above model,force rendering and control of the haptic device is decoupled from the cutting simulation loop.In the cutting simulation loop,the local model is constructed at a relatively low update rate according to the high update rate motion of the haptic device and the low update rate material removal computation.4.3Real-Time Construction of the Local ModelThe computation principle is illustrated in Fig.8,where ~N V iis the normal of the triangle’s vertex within the tooth’striangle mesh that penetrates into the virtual tool.~N T iis the normal vector of the triangle within the tooth mesh that penetrated into the virtual tool.Suppose the center point of the plane is ~X LM x p ;y p ;z p ÀÁand the normal of the plane is ~n LM n x ;n y ;n z ÀÁ.Using thedynamic ~XLM and ~n LM ,the plane can be determined.Then,~n LM ¼X n i ¼1~N T i!X n i ¼1~N T i !;,ð18Þ~X LM ¼~X V þÁd Á~n LM ;ð19Þwhere ~XV is center of the virtual tool computed in (2),Ád is the depth that the virtual tool penetrated into the tooth mesh,Ád ¼Max Ád i ðÞ;ð20Þwhere Ád i is the distance from the triangle’s vertex withinthe tooth mesh to the surface of the virtual tool and Max ðÞmeans to get the maximum value among all distances.Thedistance is computed along the negative direction of ~N V i.4.4Force Rendering in the High Update Rate LoopInside this loop,the position of the Phantom desktop is sampled and the position of the virtual tool is achieved using a motion-scaling method.Collision detection is computed between the virtual tool and the local model and virtual force can be computed.The update rate of the virtual force computation can be maintained over 1KHz because of the local model’s simplified geometry.The planar equation of the local model can be computed via the center point and normal in (18)and (19):Ax þBy þCz þD ¼0;ð21ÞA ¼n x ;B ¼n y ;C ¼n z ;ð22ÞD ¼Ax p þBy p þCz p :ð23ÞDefine the virtual proxy point ~XS x s ;y s ;z s ðÞ,which refers to the mapping position of the virtual tool to the boundary of the local model in order to compensate for the interpenetration effect of the haptic device.This position can be computed by the position of the virtual tool and the local model.x s ¼x v ÀA Áy s¼y v ÀB Áz s ¼z v ÀC Á ;8<:ð24Þwhere ~XV x v ;y v ;z v ðÞis the geometric center of the virtual tool and¼A Áx v þB Áy v þC Áz v ÀD ðÞ=A 2þB 2þC 2ÀÁ:ð25ÞThe unilateral constraint cased force model is used to compute contact force:676IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS,VOL.11,NO.6,NOVEMBER/DECEMBER2005Fig.6.Translation of the local model’sposition.Fig.7.Architecture of the multirate cutting simulationsystem.Fig.8.Parameters to compute the local model.。