Introduction

Whereas force feedback devices have reached technical qualities which allow their use in many simulation applications, the simulation of realistic deformable models remains too slow to be directly used for real-time force feedback, especially when the object geometry is complex. Research on physical deformable models has been very active for ten years[1,2] and allows us to build visually interactive models. Thanks to retina persistence a visual sensation of continuity is provided with relatively low frequencies (about 25Hz). The sense of touch is much more precise. It requires refresh frequencies ranging from 300Hz for soft objects to 10kHz for rigid contact.
In the literature, two main approaches to reach haptic real-time are developed:

• computing forces empirically [3], for example by using a force proportional to the penetration depth of the tool in the object. In this case the force is not deduced from a physical deformation.
• using a simplified physical model. The simplification can be done in two ways, either by decreasing the size of the model [4] or by doing as much pre-computation as possible [5]. In the latter case, the model can't endure any topological change. A combination of the two possibilities is also a solution with a large pre-computed part and a small simpler part whose topology can be altered [6].

Our work is placed in the context of minimally invasive surgery simulation, and more precisely in virtual hepatic resection under laparoscopy [7]. The purpose is to give the surgeon the ability to practice on a virtual patient. The simulator must be as realistic as possible both for visual and for haptic rendering, which isn't attainable with the previously mentioned approaches.
In this paper, we propose a solution based on human characteristics. It has been shown ([8]) that if the sense of touch is very precise (we can feel vibrations until 10kHz, and force variations between 30 and 300 Hz), the gesture is slower (from 1Hz for the answer to an unexpected signal and 10Hz for a reflex action). Thus, the applied forces must be refreshed at high rate, but, because it is related to user's action, their evolution is quite slow. The idea is to estimate the force between two time steps of the deformable model simulation.

First, we will quickly describe our simulator architecture. Then we will present various force extrapolation schemes, and finally we will discuss them.