Configuration-based optimization for six degree-of-freedom haptic rendering using sphere-trees

This paper presents a novel constraint-based six degree-of-freedom (6-DoF) haptic rendering algorithm for simulating both contact forces and torques between interacting rigid bodies. We represent an object using a hierarchy of spheres, i.e., a sphere-tree. Such a representation allows fast detection of multiple contacts/collisions among objects and facilitates contact constraint formulation. Given a moving graphic tool as the avatar of the haptic tool in the virtual environment, we constrain its position and orientation, i.e., its six dimensional configuration, by solving a constrained optimization problem. The constraints in the 6-D configuration space (C-space) of the graphic tool is obtained and updated through on-line mapping of the non-penetration constraint between the spheres of the graphic tool and those of the other objects in the three dimensional physical space, based on the result of collision detection. The problem is further modeled as a quadratic programming problem and solved by classic active-set methods. Our algorithm has been implemented and interfaced with a 6-DoF Phantom Premium 3.0. We demonstrate its performance in dental surgery simulations involving complex, multi-contact virtual environments. Our method enables stable operations and realistic feel of haptic sensation.