ABSTRACT HAPTIC RENDERING FOR INTERNAL CONTENT OF AN IMPLICIT OBJECT

HAPTIC RENDERING FORINTERNAL CONTENT OF AN IMPLICIT OBJECTThavida Maneewarn Blake HannafordDepartment of Electrical Engineering Department of Electrical Engineering University of Washington University of Washington/BRL /BRL Duane W. Storti Mark A. GanterDepartment of Mechanical Engineering Department of Mechanical Engineering University of Washington University of Washingtonstorti@ ganter@ABSTRACTThe haptic rendering algorithms for object with implicit surface representation are proposed. The basic rendering algorithm for a regular two-dimensional manifold is used to represent a solid volumetric object. For applications where information about the internal content of an object is concerned, the haptic rendering algorithm for internal content of an implicit object is developed. The proposed algorithm is derived from the concept of state transition via ‘port’ using manifold with boundary representation. Manifold with boundary is constructed from implicit primitives. The advantage of implicit surface representation in haptic applications has been emphasized. Both algorithms were successfully implemented in the haptic interface system and were applied to various implicit models. INTRODUCTIONBased on experience in the physical world, it is natural for humans to perceive objects as being more realistic as more of the senses are provided the feedback normally associated with interactions with a physical object. Since the goal of any CAD/CAM system is to provide a virtual representation of a part, assembly, or environment that is perceived to be as real as possible.In this paper, we concentrate on haptic feedback which provides the user with tactile sensations that either simulate interaction with the real version of the virtual object or which are designed to assist in communicating characteristics of the object. In particular, we focus on haptic rendering, the process of creating a synthetic force to replicate the reaction force associated with contacting an object.Previously introduced haptic rendering algorithms can be classified according to their reliance on standard surface representations. Zilles [13] introduced the "god-object" approach that allows polyhedral model to be rendered by restricting the surface contact point from passing through the object. Thompson [12] proposed a related algorithm that supports direct haptic rendering of NURBS surfaces.Some algorithms employ a local approximation of the surface to reduce computation time. Adachi [1] and Mark [4] use a slowly changing planar approximation as an intermediate representation of a surface. Zilles [13], Morgenbesser and Srinivasan [7] and Ruspini [10] applied a technique called ‘force shading’, analogous to Phong shading in computer graphics, to render surface curvature by interpolating surface normal vectors of polyhedral facets.Methods that do not rely directly on surface representation of a geometric model have also been suggested, such as a potential field method due to Massie and Salisbury [5] and a voxel-based method [2] that determines haptic interaction forces based on the current position of the user interaction point. Voxel-based methods are suited for rendering large volumetric objects, but they are memory intensive and may not be applied to objects with feature sizes comparable to voxel sizes.Implicit surface representationImplicit surfaces are two-dimensional, geometric shapes that exist in three-dimensional space. The surface is implicitly described by an analytic function S(p) as S(p) = 0. The use of implicit surfaces in computer graphics was initially introduced in scan-line rendering of quadric surfaces (Davis et al. [3] and Mittelman [6]). The implicit function characterizes volume as well as surface, since value of S(p) indicates the proximity between point P and the surface. Implicit functions can not only be used to describe basic geometric shapes. The model of complicate shapes can be achieved by applying appropriate blending operations to basic implicit primitive [9]. The use of implicit surface representation in geometric modeling has increased due to its unique properties. These properties include simple collision detection between a point and the surface that can be achieved from function evaluation. If S(p)<0, point P is enclosed inside the surface. If S(p)>0, point P is outside the surface and if S(p) = 0, point P is on the surface. The gradient of S(p) is proportional to the surface normal vector pointing in outward direction. The gradient of S(p) can be described by)/)(,/)(,/)(()(zpSypSxpSpS∂∂∂∂∂∂=∇The surface normal vector indicates the direction of a responsive force from the implicit model. The gradient of an implicit equation can be calculated both analytically and numerically. Anotherinteresting property is that the inside/outside can simply be reversed by sign inversion of S(p). This property is very useful in a rendering process for some class of objects, which will be discussed in the following section.The haptic rendering of implicit surfaces was first introduced by Salisbury and Tar[13]. They proposed a method for rendering solid objects represented by regular two-manifold and extended to address the specific problem of rendering finites solid cones or cylinders created from intersecting half-spaces with an implicit solid. However,their method only applies to the surfaces enclosing solid volumetric objects regardless of the internal content that may be confined inside the surfaces.In geometric modeling, the non-regularized manifolds such as ‘manifold with boundary’ and ‘non-manifold’ are also concerned. The non-regularized manifold description can be applied to objects that consist of non-uniform or composite materials, mostly used in biomedical or electronics applications. Manifold with boundary is a surface that locally approximated either by a disk or by a half-disk. The manifold with boundary represents object that contains openings or dangling edges. The haptic rendering technique proposed in this paper can be applied to more general classes of objects in addition to the solid volumetric object described by the two-manifold. We would like to emphasize the advantage of implicit representation in haptic rendering where the object interior can be haptically displayed. One example described here is to simulate the effect of having non-solid substance confined in the space enclosed by implicit surfaces, such as liquid in a cup. The algorithm is very simple to implement and is not computationally expensive, which make it appropriate to be performed in real-time. The model can be well described by a small number of implicit equations. Thus only small memory space is required in this method unlike in voxel-based method or in polyhedral model-based method.Figure 1.1 A cup that contains liquid inside can be described by manifold with boundary constructed from implicit primitives.IMPLICIT HAPTIC RENDERING ALGORITHMThe haptic rendering algorithm for objects defined by implicit surfaces can be described in two parts, the rendering algorithms for a regular two-dimensional manifold and the rendering algorithm for internal content of an object using manifold with boundary description.Although the rendering algorithm for regular two-dimensional manifold proposed here has several aspects that are quite similar to previous work by Salisbury and Tar [11], our implementation details are different from theirs in various points. For instance, the Lagrangian multiplier is used, in the method introduced by Salisbury and Tar, inorder to find the surface contact point which minimize distance between the seed point (the probe point that is inside the surface for the first time) and the point on the surface. The problem of using the closest point on the surface instead of the actual surface crossing point can be illustrated in figure (2.1). When the probe point approaches the convex surface, the surface contact point is established at )(k P o ′which is the nearest point from )(k P instead of the actual surface crossing point at )(k P o . The responsive force that is calculated from )(k P o ′ may be incorrect and it may cause an undesirable effect. In our algorithm, the binary search is used to establish the surface contact point at the crossing point between a line segment and the surface, which will provide better accuracy. Details of our haptic rendering algorithm for a regular two-dimension manifold will be described in the following section.Figure 2.1 A situation where surface contact point can be incorrectly established at the closest point between the seed point and the implicit surface.Rendering a regular two-dimensional manifoldThe rendering algorithm for regular two-dimensionalmanifold is an important basis for more complicate rendering algorithm that will be presented in the next section. Two-dimension manifold is often used to describe the surface enclosing solid objects where the point-set inside the surface is entirely separate from the outside space. When the probe point is contacting with solid objects,the responsive force is a force that prevents a probe point from penetrating into the surface. The responsive force is defined by an equation )())()((k BV k P k P K f i o −−=, where K representsstiffness and B represents damping of a solid object. o P (k) is the surface contact point and i P (k) is the interaction point inside the surface. V(k) is velocity of the probe point.Figure 2.2 A responsive force resulted from contacting the surface)(k P o )(k P i ))()((BV k P k P K f i o −−=)(k P o )(k P o )(k P )1(−k PThe algorithm for calculating the responsive force can be described as followed. )(k P o and )(k P i , which are used in force calculation, are found from algorithm I and II, depends on the condition of the current position (P(k)) and the previous position (P(k-1)) of the probe point input. The collision occurs when the probe point is inside the surface for the first time which means P(k-1) is outside the surface (0))1((>−k P S ) and P(k) is inside the surface (0))((<k P S ). When the collision occurs, algorithm I is used to define the new interaction points )(k P o and )(k P i . After the collision and if P(k) still be inside the surface, algorithm II will be used to define the next interaction points. Note that, in the special case where point )(k P o which is calculated from algorithm II is too far outside the surface ( ε>))((k P S o ), the new interaction points will have to be recalculated as the surface crossing point on the line segment from )(k P o to )(k P i using algorithm I.Algorithm I is performed when the probe point P(k) collideswith the surface for the first time. The surface contact point )(k P o is established at the point where the surface crosses the line segment between point P(k) and P(k-1). The implicit function S(p) evaluates zero at )(k P o . Since the update rate in haptic rendering process is very high, only a small movement is permitted within one interval (i.e.distance between P(k) and P(k-1) is small). The surface crossing point where 0))((=k P S o can be found using binary search along the segment between P(k) and P(k-1). (Figure 2.3)Figure 2.3. The surface contact point )(k P o is found on the line segment between P(k) and P(k-1)After the surface contact point )(k P o is founded, theinteraction point inside the surface )(k P i is established by mapping the probe point input P(k) onto the surface normal vector at )(k P o so that the direction of force vector calculated from )(k P o and )(k P i is appropriately defined with respect to the implicit surface.Figure 2.4 The interaction point inside the surface )(k P i is established from P(k) and the surface normal at )(k P o It is also important to make sure that )(k P o is not inside the surface.Therefore, when implicit function evaluated at )(k P o is less than zero, )(k P o has to be moved outside the surface by a very small distance (specified by α as shown in algorithm I) in the direction of surface normal evaluated at )(k P o .Algorithm I: Find the new interaction point%% Find Po(k)If P(k) is inside the object for the first time%% Binary search for the interaction point on the surface DoPo(k) = (P(k)+P(k-1))/2;Until S(Po(k)) < ε or MAX_N iter.%% Find Pi(k)%% N(P(k)) is the surface normal vector at point P(k)))()(()(k P k P k vP o −=))())(())(())(())(()((()()(k vP k P N k P N k P N k P N k vP k P k P o o o o i −⋅⋅+=%%check if Po(k) is inside the surface then modify Po(k)if S(Po(k)) < 0))(())(())(()()(k P N k P N k P S k P k P o o o o o α−=return Po(k) and Pi(k)When the probe point is still inside a surface after the first collision with the surface, Algorithm II is used to move the interaction point on the surface from the previous state )1(−k P o to the new position on the surface based on the movement of probe point P(k-1) to P(k). The movement of the probe point inside the surface is used to calculated the interaction point )(k P o , by mapping the vector from P(k-1) to P(k)onto the tangent plane at )1(−k P o . )(k P o is modified by a small distance in the direction of the surface normal evaluated at )1(−k P o to guarantee that )(k P o will not be inside the surface.(k P P(k)P(k-1))(k P i is also calculated in the same way as in Algorithm I so that the direction of reactive force is defined correctly with respect to the implicit surface.Figure 2.5 The surface contact point )(k P o is established based on the movement of probe point inside the surfaceAlgorithm II: Find the next interaction point %% Find Po(k)))1()(()(−−=k P k P k V )))1(())1(()1((())1(())1(())1(())1(()()(()1()(−−−⋅−−−⋅−−⋅−+−=k P N k P N k P S k P N k P N k P N k P N k V k V k P k P o o o o o o o o o α%% Find Pi(k)))()(()(k P k P k vP o −=))())(())(())(())(()((()()(k vP k P N k P N k P N k P N k vP k P k P o o o o i −⋅⋅+=return Po(k) and Pi(k)Rendering the internal content of an implicit objectNormally, a solid model is represented by a closed surfaceseparated the outside space from the inside. In the regular solid model,no information about what contains inside the surface is available. In practice, internal content of an object often provides significant amount of useful information, such as property of composite material that contained inside the object or a special form of internal structure of the object. The ability to sense internal content of an object becomes more important in several applications especially in biomedical and electronics applications.The internal content of geometric model can be described bymanifold with boundary. The reason that manifolds with boundary are used is so that the transition between internal and external space can only occur within the restrictive region. Using the manifold with boundary representation, the opening region bounded by the manifold boundary is defined as ‘port’ where the transition between spaces takes place. In other word, internal content of an object can only be displayed after the probe point has entered a defined region called ‘port’. The concept of traveling through ‘port’ allows interior and exterior force of the geometric model to be distinctively displayed via state transition.Manifold with boundary is constructed from implicitsurface primitives. Implicit surface representation is required because implicit function posses unique properties that the surface can be simply reversed by a sign inversion and inside/outside can be easily determined by evaluating an implicit function. The exterior force is defined to simulate the responsive force from contacting the exterior surface of a geometric model. Haptic rendering process for the exterior surface can be performed in the same way as for a regular two-dimensional manifold surface described in the previous section.Internal content can be displayed by various techniques depends on material property or the arrangement of the internal structure. For example, viscous effect can be used to represent non-solid viscous substance that confined inside the implicit surface or a spatial periodic oscillation can be used to represent a grid-like structure of internal material.As a basic example, the manifold with boundaryconstructed from an implicit half-plane and a sphere is demonstrated (as shown in figure 2.6). The state transition conditions for these two implicit primitives are shown in Table I. Four distinctive regions are defined in the state transition table. State transition is only performed when the probe point is traveling through the ‘port’ which defined as an intersect region between a half-plane and a sphere. If the probe point is inside both surfaces (a sphere and a half-plane), the exterior surface of a sphere will be displayed unless the probe point has entered ‘port’ in the previous state. In that case, the internal content of a sphere will be displayed instead. In this example, a sphere container that contains non-solid viscous substance is simulated. The exterior force is calculated as the responsive force from contacting the outer shell of a sphere container. The interior force is a combination of the viscous resisting force and contacting force from the surface interior.The viscous resisting force is proportional to velocity of the probe point, which can be described by the equation )(k V B F v v −= , wherev B is the viscous coefficient of a simulated non-solid substance andV(k) is the probe point velocity. The contacting force from the surface interior is calculated in the same way as the responsive force from contacting a regular two-manifold, except that the reversed implicit function of sphere is used instead. At each servo loop, collision detection between the probe point and implicit surfaces is performed for each surface in the same way as described in the method for a regular two-manifold. The proper responsive force is calculated based on the status of the probe point as shown in the state transition table.Figure 2.6. Manifold with boundary that represents a sphere container, constructed from a sphere and a half-plane.V (k)P(k-1)State Sphere Half-plane1IN IN If (the previous state is 2)ENTER_PORTElseIf (ENTER_PORT has occurred)Find a viscous resisting forceElseFind a contact force of sphere-exterior surface2IN OUT If (the previous state is 1)EXIT_PORT3OUT IN If (ENTER_PORT has occurred)Find a contact force of reversed -sphere surfaceElseF = 04OUT OUTF = 0Table I. State transition table for 2 implicit primitivesMore complex example is illustrated in figure 2.7. Compositematerial is confined inside a cylinder tube. In the case where n implicitsurfaces are required to describe a model, 2n distinctive regions areintroduced. Different material properties will be displayed as the probepoint passes through each layer. Although state transition table for thismodel will be much more complicate than in the previous example, it isbased on the concept where conditions can be tested iteratively. (i.e. theprobe point cannot enter port 2,3,4 unless it has entered port 1,therefore it is not necessary to test every conditions at each servo loop.)The rendering algorithm for complex internal structure within ageometrical model can be simplified by applying this concept of stateFigure 2.7 A cylinder tube that contains composite material can bemodeled using state transition via ‘port’ concept.State transition via ‘port’ can also be applied to an applicationwhere the ‘flow direction’ is concerned. The flow direction means thatonly a certain direction is allowed to travel within a defined space.According to the state transition via ‘port’ paradigm, arbitrary regioncan not be accessed unless the probe point has traveled through theprior region defined by a sequence of ‘port’.Objects with complicate shape can be modeled bothexternally and internally by applying constructing techniques to basicimplicit primitives as we has discussed earlier. The haptic renderingalgorithm proposed here emphasizes the advantage of implicit surfacerepresentation in haptic applications where the object interior can behaptically displayed as well as the exterior surface.HAPTIC INTERFACE SYSTEMThe implicit haptic rendering algorithm described above wasimplemented in the haptic interface system at Biorobotics LaboratoryUniversity of Washington using Excalibur, the 3 degree-of-freedomhaptic linear haptic display device (shown in figure 3.1). TheExcalibur was designed and manufactured by HapticTechnologies,Inc. of Seattle, WA. The user can provide the positional input bymoving the handgrip that connects to light linkage elements. The 3-dimensional force feedback is generated from motors mounted on thebase of the mechanism. The Excalibur is using steel cabletransmission that enables high forces and high rigidity in the three-axis translation motions. The workspace of the device is300x300x200 mm3 with the maximum force of 100 N and theposition resolution of 0.008 mm.The haptic rendering algorithm is implemented on a 266 MHzPentium II processor PC that controlled the Excalibur in real-timewith the update rate of 1 KHz.Figure 3.1. Excalibur, the 3 degree-of-freedom haptic linear hapticdisplay deviceSeveral examples of implicit models were hapticallyrendered using the methods described in the previous section. Theseexamples include1.Basic shapes such as sphere, ellipsoid, cube and heart.2.Metaballs: 2-5 blended spheres.3.CSG: union, intersection and difference of basic primitives suchas ellipsoid and cube.4.Implicit container: A sphere container with the opening on thetop (Figure 3) that contains liquid with adjustable viscouscoefficient.The user was able to correctly recognize the shape ofrendered objects by using tactile senses alone without graphicalinterface. The proposed algorithm is simple and computationallyinexpensive enough to be performed within 1 millisecond timeinterval. However, undesirable oscillation sometimes occurs in highconvex curvature region. Applying the appropriate adjustment to thenormal vector calculation procedure may alleviate this problem. Thebounding box or localization technique should be implemented in thealgorithm so that more complex model which consists of a largenumber of primitives can be haptically displayed in real-time.CONCLUSIONSThe implicit surface representation is well suited for hapticapplications, especially for modeling the object consisting of quadricprimitives. Constructing techniques in geometric modeling allow complex shapes to be haptically displayed as well as basic shapes. The haptic rendering methods for a regular two-dimensional manifold and for the internal content of an object were presented in this paper. The properties of implicit surface representation are suitable to be applied to haptic rending process for both classes of objects. The new concept of state transition via ‘port’ model is introduced so that the internal content of an object can be rendered haptically. Haptic rendering of various implicit models was demonstrated including the combination of geometric primitives and the model of implicit container that contains non-solid viscous substance. Unfortunately, implicit surface representation is still not widely used in geometric modeling compare to other well-established methods such as parametric surface representation or boundary representation. This factor may prevent the implicit haptic rendering technique to be applied to the existing geometric modeling system. Hopefully, as the attention of implicit surface modeling in graphics increases, it will consequently bring more interest into implicit haptic rendering applications in the near future. 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