Collision Detection Between Convex Objects Using Pseudodistance and Unconstrained Optimization

The problem of collision detection plays an important role in many fields of science and engineering. This article presents a collision detection method for general convex objects bounded by pieces of implicit surfaces. There are two key ideas that underlie our method: one is the introduction of a new kind of pseudodistance, called the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-distance, for implicitly represented convex objects which has the desired properties of convexity and square differentiability; the other is the use of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-distance functions to construct a virtual potential field in the real space, so that the problem of collision detection can be reduced to a problem of unconstrained convex optimization. The method is extended and applied to detect whether two objects collide when they are moving continuously along linearly translational trajectories, which is a special case of one of the continuous collision detection subproblems. We have implemented collision detection algorithms in C++ and conducted a large number of experiments, with test examples involving objects modeled by planar, quadric, superquadric, superellipsoidal, and hyperquadric surfaces, as well as pieces of them, in both stationary and linearly translational moving states. The experimental results show that our method has good performance and it is computationally efficient and widely applicable.