We present a novel geometric data structure for approximate collision detection at haptic rates between rigid objects. Our data structure, which we call inner sphere trees, supports both proximity queries and the penetration volume; the latter is related to the water displacement of the overlapping region and, thus, corresponds to a physically motivated force. Moreover, we present a multi-threaded version of the penetration volume computation for time-critical haptic rendering that is based on separation lists and the novel notion of expected overlapping volumes. Finally, we show how to use the penetration volume to compute continuous contact forces and torques that enable a stable rendering of 6-DOF penalty-based distributed contacts. The main idea of our new data structure is to bound the object from the inside with a bounding volume hierarchy, which can be built based on dense sphere packings. In order to construct an efficient hierarchy, we propose to use an AI clustering algorithm, which we extend and adapt here. The results show performance at haptic rates both for proximity and penetration volume queries for models consisting of hundreds of thousands of polygons.

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