Deformation

Deformation has been throughly studied in computer graphics, but still remains very challenging in computer haptics. Collision and force computation with deformable objects is not in the focus of this work, however, this section provides a brief overview of the most relevant methods. For deeper insights, the reader might consider the reviews by Teschner et al. [TKZ+_{05] and Nealen et al. [NMK}+_06].

In general, collision computation, force rendering and deformation are treated as separate problems computed in independent threads that share critical information. As introduced in Section 2.3.1, particle or mass-spring systems [CKB03] and tetrahedron meshes [SSB13] are common for rendering deformation. In both cases, nodes (particles or vertices) are interconnected with one another, and their displacement upon external forces is computed. Tetrahedron meshes are common when the Finite Element Method (FEM) is used to solve deformation equations (boundary value problems) defined in continuous but discrete mass cells (usually, the tetrahedra themselves). With mass-spring systems, the mass is concentrated at the node points and the forces are propagated along the linking spring edges. While FEM produces more accurate deformations, mass-spring systems are computationally less expensive; however, constant volume deformation is difficult to model with them.

Independently of the used methods, it is common to speed up the processing of equa- tion systems that lead to the deformed configuration with several shortcuts; for instance, the Sherman-Morrison formula for incrementally updating the inverse of the equation system, pre-computed basis deformation functions [JP01], modified Gauss-Seidel algo- rithms that converge to the solution of the chosen contact model very fast given an error tolerance [DDKA06], etc.

Intermediate representations have also been investigated as a method to support haptic update rates with deformable objects, as done by Garre et al. [GO09]. In their work, the deformable object or tool that interacts with the environment has a rigid part, the handle, which is connected to the haptic device. The coupling force between both

parts is linearized and fed to a virtual coupling model that renders the forces perceived by the user. For updating the linearized coupling force model, the velocity values of the handle, the tool, and the environment need to be solved from a nonlinear complementarity problem. During the simulation, the framework runs in two separate loops; a visual loop simulates the deformations of the tool using FEM at 30 Hz, whereas the haptic loop evaluates the updated linearized coupling force with the current handle state and delivers the subsequent forces at 1 kHz to the user who is moving the device. Peterlik et al. [PND+_{11] presented a related approach with an intermediate representation in} the context of surgical simulations with haptic feedback. In this case, the intermediate representation is a system of compliant mechanisms that is shared by the slow dynamics loop, which updates the model according to the deformations of the objects using FEM, and the fast haptics loop, which evaluates the contact force on the model. In other words, the intermediate representation extracts from the FEM model used to compute deformations the mechanical compliance in contact used to render forces.

One elegant six-DoF haptic rendering method which covers all the necessary processes for interaction with deformable objects was presented by Barbič and James [BJ08]. The method builds up on the VPS algorithm [MPT99], which computes contact forces between distance fields and point clouds. These data structures are embedded in reduced FEM models [SSB13], which receive as input values the computed contact forces. Since the point coordinates of the point cloud are anchored in the FEM model, their deformed positions can be obtained straightforwardly. In the case of the distance fields, a low resolution deformable point cloud is embedded during data structure generation. The deformed point coordinates of this auxiliary point cloud are used to locally update the values of the distance field computed for the resting configuration. A similar approach is followed by Chan et al. [CBS13], in this case with intensity fields embedded in FEM tetrahedral meshes. In oder to obtain the field values of the deformed environment, first, the barycentric coordinates of the queried point in the deformed tetrahedron where it lies are computed. Substituting these coordinates in the rest configuration leads to the equivalent material point to query in the (undeformed) intensity field.

Finally, related to but beyond deformation, some simulations consider object topol- ogy change in scenarios that cover material substraction or cutting. These are usual in medical simulations, in which tissue dissection is common, either with organs or bones. In this respect, Courtecuisse et al. [CJA+_{10] presented a realtime haptic simulation of} soft tissue manipulations, including cutting. The method uses co-rotational FEM models that handle large displacements and geometrical non-linearities. When a tissue is cut, its mesh is modified, as well as the FEM model and the matrix elements in the equations of motion. To that purpose, the deformable model is meshed into set of volume elements which are connected together by a topological map. The state and model matrices are

expanded with element subtraction and subdivision operations, and the efficient update of inverse matrices necessary for stepping forward the dynamic state is tackled using the Sherman-Morrison method, among others. On the other hand, Wang et al. [WCW+_12] presented an impulse-based haptic rendering framework for the specific scenario of dental bone-burring with a rapidly rotating spherical spindle. The environment (e. g., tooth or bone) is represented with a mesh, whereas the tool (i. e., the spherical burring spindle) is modeled with points on its cutting edges. Rays are shot from the tool center to the points on the edges to detect point and triangle pairs which are colliding. The mesh is reconstructed replacing all vertices of burred triangles to be on the boundary of the spherical spindle.

However, bone-burring scenarios have been handled more commonly using voxelmaps, due to their efficient data structure update capabilities and minimal to non-existent topol- ogy changes upon material subtraction. Additionally, this way, patient-specific data structures can be obtained from Magnetic Resonance Imaging (MRI) Computer Tomog- raphy (CT) models. Zheng et al. [ZLSWF13] presented a training simulator for dental surgeries using a re-implementation of the VPS for GPUs. A density field embedded in a voxelmap is created from CTs for the teeth, recreating different tissue hardness values. Since drilling is simulated, the field values change during the interaction in the burred regions, and the visualization model is constantly updated using the marching cubes technique [LC87]. Specific spindle tools are supported, placing on their cutting edges the points used for collision and force calculation (i. e., pointshell points). The rotational speed and angle of incidence of the spindle are considered in the force and torque com- putation, respectively, improving training realism. Kim and Park [KP09] presented a similar penalty-based approach, but using distance fields for the tool and intensity fields and point clouds for the teeth. The authors additionally defined a bone removal rate function to gradually subtract tissue elements on contact.

Other works have focused on proper force rendering methods in bone-burring sce- narios using volume-based data structures. Agus et al. [AGG+_{03] derived an analytical} contact force and bone erosion model based on the Hertz’s elastic contact theory [Her96]; geometric and elastic parameters are taken into account and the model is discretized for voxelmaps. Wu et al. [WYW+_{09] discovered and applied a linear relationship model for} resistance force and forward velocity.