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Six-degree-of-freedom (6-DOF) haptic rendering is challenging when multi-region contacts occur between the graphic avatar of a haptic tool operated by a human user, which we call the graphic tool, and deformable objects. In this paper, we introduce a novel approach for deformation modeling based on a spring-sphere tree representation of deformable objects and a configuration-based constrained optimization method for determining the 6-dimensional configuration of the graphic tool and the contact force/torque response to the tool. This method conducts collision detection, deformation computation, and tool configuration optimization very efficiently based on the spring-sphere tree model, avoids inter-penetration, and maintains stability of haptic display without using virtual coupling. Experiments on typical dental operations are carried out to validate the efficiency and stability of the proposed method. The update rate of the haptic simulation loop is maintained at ~1kHz.

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