## Abstract

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 M^{6} in the 12-dimensional space R^{12} 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 M^{6} 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 M^{6} 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}.

Original language | English (US) |
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Pages (from-to) | 425-443 |

Number of pages | 19 |

Journal | Computer Aided Geometric Design |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - May 2009 |

## Keywords

- Distance function
- Geometric optimization
- Gliding motions
- Kinematics
- Penetration depth

## ASJC Scopus subject areas

- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design