**3.1.4 Modeling friction**

Friction is important for providing surface information during object exploration, and it is fundamental in the context of grasping. Friction is modeled by means of the Coulomb's friction law that is based on two states: stick and slip. In stick state the norm of the tangential component of the force between two objects is less than the product of the norm of the normal force by the static friction coefficient. In this case there is no relative motion. When the tangent component has a norm larger than the proportional normal force the slip occurs. In this case the effective tangential force is directed against relative motion and proportional to the normal component of the force by the dynamic friction coefficient.

In the basic case of mass-less rigid proxy, as in the god-object algorithm, Coulomb friction can be implemented by means of the friction cone algorithm (Melder & Harwin, 2004). In this case the force is obtained by a spring connected between proxy and handle. Without friction the force is directed along the normal of the surface based on the position of the proxy. In the friction cone model the stick state keeps the proxy in the previous position producing a tangential force up to the level of tangential force that makes the proxy enter the slip state.

The friction model in rigid body simulation can be performed with reduced precision by means of a sequential resolution of frictional contacts. In this case the friction force is applied as an external force that is added to other methods for contact resolution. More sophisticated models take into account the friction model in the integration step extending the LCP model. Specifically the friction cone is represented in the equation as a k-sided polygonal cone, at the cost of increased complexity of the system to be solved. Alternatively Durez et al. integrated friction cone in deformable haptics using Gauss-Seidel algorithm improving performance and precision of the friction model.

## **3.1.5 Soft fingers**

Deformable bodies for contact allow us to introduce an important aspect for the rendering of direct interaction: the soft modeling of fingertips. Barbagli et. al (F Barbagli, A Frisoli, K Salisbury, & M Bergamasco, 2004) discussed fingertip contact deformation models and measured several in-vivo characteristics for comparing them with the models. In particular the indentation displacement, contact area and friction coefficient. These measurements allowed them to design a soft-finger 4 DOF proxy algorithm that took into account torsional friction based on applied pressure (Antonio Frisoli, Federico Barbagli, Ruffaldi, Massimo Bergamasco, & Ken Salisbury, 2006). The investigation on the model of human fingertip can be applied on the haptic rendering of soft fingers when they interact with rigid and deformable objects. In particular Ciocarlie et al. (Ciocarlie, Lackner, & Allen, 2007) presented a method for computing the soft finger contacts based on local geometries and object curvatures.

## **3.1.6 High frequency contact**

58 Haptics Rendering and Applications

example, the constraint model is applied using filtering of the motion inside a Conjugate Gradient method (Baraff & Witkin, 1998). Duriez et al. (Duriez, Andriot, & Kheddar, 2004), instead, adopted a contact model that employs Signorini's law for quite convincing FEM model supporting contacts between deformable objects. In particular the contact is resolved by equating the gap with a combination of a post contact gap and the projection of the two displacements along the contact normal. This equation is then expressed in terms of the

Friction is important for providing surface information during object exploration, and it is fundamental in the context of grasping. Friction is modeled by means of the Coulomb's friction law that is based on two states: stick and slip. In stick state the norm of the tangential component of the force between two objects is less than the product of the norm of the normal force by the static friction coefficient. In this case there is no relative motion. When the tangent component has a norm larger than the proportional normal force the slip occurs. In this case the effective tangential force is directed against relative motion and proportional to the normal component of the force by the dynamic friction

In the basic case of mass-less rigid proxy, as in the god-object algorithm, Coulomb friction can be implemented by means of the friction cone algorithm (Melder & Harwin, 2004). In this case the force is obtained by a spring connected between proxy and handle. Without friction the force is directed along the normal of the surface based on the position of the proxy. In the friction cone model the stick state keeps the proxy in the previous position producing a tangential force up to the level of tangential force that makes the proxy enter the slip state.

The friction model in rigid body simulation can be performed with reduced precision by means of a sequential resolution of frictional contacts. In this case the friction force is applied as an external force that is added to other methods for contact resolution. More sophisticated models take into account the friction model in the integration step extending the LCP model. Specifically the friction cone is represented in the equation as a k-sided polygonal cone, at the cost of increased complexity of the system to be solved. Alternatively Durez et al. integrated friction cone in deformable haptics using Gauss-Seidel algorithm

Deformable bodies for contact allow us to introduce an important aspect for the rendering of direct interaction: the soft modeling of fingertips. Barbagli et. al (F Barbagli, A Frisoli, K Salisbury, & M Bergamasco, 2004) discussed fingertip contact deformation models and measured several in-vivo characteristics for comparing them with the models. In particular the indentation displacement, contact area and friction coefficient. These measurements allowed them to design a soft-finger 4 DOF proxy algorithm that took into account torsional friction based on applied pressure (Antonio Frisoli, Federico Barbagli, Ruffaldi, Massimo Bergamasco, & Ken Salisbury, 2006). The investigation on the model of human fingertip can be applied on the haptic rendering of soft fingers when they interact with rigid and deformable objects. In particular Ciocarlie et al. (Ciocarlie, Lackner, & Allen, 2007) presented

improving performance and precision of the friction model.

exchanged contact forces and resolved in the general deformable FEM framework.

**3.1.4 Modeling friction** 

coefficient.

**3.1.5 Soft fingers** 

The position control approach for rendering first contact is not able to represent high frequency transients that characterize stiff materials. A possible solution to this problem has been addressed by event-based haptics (Kuchenbecker, Fiene, & Niemeyer, 2005) in which the first instants of contact are performed in open loop by superimposing a previously recorded force profile.
