Generalized penetration depth computation based on kinematical geometry Original Research Article

The generalized penetration depth PD of two overlapping bodies image and image is the distance between the given colliding position of image and the closest collision-free Euclidean copy image to image 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 image into consideration. We use a kinematic mapping which maps rigid body displacements to points of a 6-dimensional manifold image in the 12-dimensional space image 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 image which correspond to colliding configurations. Based on the theory of gliding motions, the closest point with respect to the metric S (image) 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 image induced by S (image). Moreover we introduce two methods for the computation of a collision-free initial guess and give a physical interpretation of image and image.