**3.1.2 Rigid objects**

56 Haptics Rendering and Applications

physical one. The former describes the geometrical representation of the shape of the proxy and the objects touched, while the latter comprises the effects of deformation and material properties like friction. In terms of temporal scale it is possible to organize contact distinguishing between first impact and then continuous contact, having two different time

At beginning of haptic rendering research, contact has been modeled by representing the proxy as a single point like in the reference god-object algorithm (Zilles & J. Salisbury, 1995) or as a rigid sphere (Ruspini et al., 1997). In both cases the interaction between the proxy and the objects is based on geometrical considerations with the objective of avoiding penetration. The proxy is mass-less, and it produces a force on the handle proportional to the material's stiffness without being affected by such force. This model is effective for rendering tool proxies that are rigid, with the advantage of high performance, requiring only point or sphere contact with object geometry. This model contains several simplifications that allow discussing the later improvements. First, both the proxy and the object are considered rigid in geometrical terms, while for realistic contact it will be necessary to represent soft fingers and deformable bodies. Second, the contact has no friction, an aspect that can be integrated with or without a complete physic simulation of the proxy. Third, the feedback has only a force component, while contact for grasping requires a torsion component. Fourth, contact is quasi static because there is low frequency contact

There is anyway a general result that has been applied by later approaches with different geometry models and physical properties: the proxy is constrained to move over the surface of the object without penetration while the haptic handle pulls it around. The other general result is the importance of a collision detection method that allows to identify or to predict the intersection of the proxy with the object. For a review on the topic see (Teschner et al., 2005). As in this case it is not necessary that both proxy and object have the same geometrical representation, it is instead more usual the case of adopting an asymmetric

Contacts between objects can be represented with few entities that do not depend on the geometrical representation and the collision detection algorithm. For two contacting objects A and B we identify the two points P and Q that are computed as the innermost points of collision respectively on the two objects. The normal of contacting surfaces can be a general n vector not necessarily directed along QP, and assumed to be toward the inner part of A. In addition, the velocities of the two points are provided, all in world coordinates. The Signorini law of contact expresses the contact of two generic objects by means of two functions: the first is the stress exerted on an object at a given point, and the second is the gap, or penetration depth, that is the projection of the QP vector on the normal. For the assumptions before a positive gap means that the two objects are penetrating. The Signorini model states that, when the contact is resolved, the gap is zero and the object B exerts a pressure toward the object A at point P, or the gap is negative and there is no pressure. When pressure is exerted it can be represented by a force directed along the normal n.

scheme, knowing that the proxy object is under the control of the user.

scales and physical modeling.

transient.

**3.1.1 Contact model** 

Objects for haptic interaction can be represented in several ways like implicit functions describing the surface (K Salisbury & Tar, 1997), volumetric objects based on voxels, or distance fields, but the most common are triangulated meshes that allow to rely on proven techniques from the fields of simulation and computer graphics. Due to the timing constraints of haptic rendering these representations can take advantage of boundary representation for collision detection or hierarchical representation of the object for reducing the computational effort. An interesting example is the technique of sensation preserving simplification by Otaduy (Otaduy & Lin, 2003) in which an object is represented by a hierarchy of variations of the object, each more detailed than the parent. Every level is represented by an aggregation of convex parts. In this approach the proxy is also an aggregate of convex parts, while collision detection is performed at a given level by comparing pairs of convex elements using the effective GJK algorithm (Gilbert, Johnson, & Keerthi, 1988). The sensation preservation is taken into account when the algorithm has to decide if it is necessary to descend into the hierarchy or to compute the force feedback at the current level. Surface properties of the pair are used to evaluate if the additional details can provide more sensation information or they are not influent.
