Generalized penetration depth computation based on kinematical geometry

The generalized penetration depth PD of two overlapping bodies X and Y is the distance between the given colliding position of X and the closest collision-free Euclidean copy X^ε to X according to a distance metric. We present geometric optimization algorithms for the computation of PD with respect to an object-oriented metric S which takes the mass distribution of the moving body X into consideration. We use a kinematic mapping which maps rigid body displacements to points of a 6-dimensional manifold M⁶ in the 12-dimensional space R¹² of affine mappings equipped with S. We formulate PD as the solution of the constrained minimization problem of finding the closest point on the boundary of the set of all points of M⁶ which correspond to colliding configurations. Based on the theory of gliding motions, the closest point with respect to the metric S (⇒ PD_S) can be computed with an adapted projected gradient algorithm. We also present an algorithm for the computation of the closest point with respect to the geodesic metric G of M⁶ induced by S (⇒ PD_G). Moreover we introduce two methods for the computation of a collision-free initial guess and give a physical interpretation of PD_S and PD_G.