## Abstract

Penetration depth (PD) is a distance measure that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translationalPD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalizedPD . When an object undergoes a rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object, under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is the same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both of the objects are non-convex, we treat them as a combination of the above two cases and present an algorithm that computes a lower and an upper bound on the generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6-dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of rigid robots undergoing translational and rotational motion in a plane or in 3D space. In particular, we use generalized PD computation for performing C-obstacle query and checking path non-existence.

Original language | English |
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Pages (from-to) | 625-638 |

Number of pages | 14 |

Journal | CAD Computer Aided Design |

Volume | 39 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2007 |

### Bibliographical note

Funding Information:This project was supported in part by ARO Contracts DAAD19-02-1-0390 and W911NF-04-1-0088, NSF awards 0400134 and 0118743, ONR Contract N00014-01-1-0496, DARPA/RDECOM Contract N61339-04-C-0043 and Intel. Young J. Kim was supported in part by the grant 2004-205-D00168 of KRF, the STAR program of MOST, and the ITRC program. We would also like to thank the anonymous reviewers for their helpful comments.

## Keywords

- Penetration depth