Efficient motion planning using generalized penetration depth computation

Motion planning is a fundamental problem in robotics and also arises in other applications including virtual prototyping, navigation, animation and computational structural biology. It has been extensively studied for more than three decades, though most practical algorithms are based on randomized sampling. In this dissertation, we address two main issues that arise with respect to these algorithms: (1) there are no good practical approaches to check for path non-existence even for low degree-of-freedom (DOF) robots; (2) the performance of sampling-based planners can degrade if the free space of a robot has narrow passages. In order to develop effective algorithms to deal with these problems, we use the concept of penetration depth (PD) computation. By quantifying the extent of the intersection between overlapping models (e.g. a robot and an obstacle), PD can provide a distance measure for the configuration space obstacle (C-obstacle). We extend the prior notion of translational PD to generalized PD, which takes into account translational as well as rotational motion to separate two overlapping models. Moreover, we formulate generalized PD computation based on appropriate model-dependent metrics and present two algorithms based on convex decomposition and local optimization. We highlight the efficiency and robustness of our PD algorithms on many complex 3D models. Based on generalized PD computation, we present the first set of practical algorithms for low DOF complete motion planning. Moreover, we use generalized PD computation to develop a retraction-based planner to effectively generate samples in narrow passages for rigid robots. The effectiveness of the resulting planner is shown by alpha puzzle benchmark and part disassembly benchmarks in virtual prototyping

Efficient computation of discrete Voronoi diagram and homotopy-preserving simplified medial axis of a 3d polyhedron

The Voronoi diagram is a fundamental geometric data structure and has been well studied in computational geometry and related areas. A Voronoi diagram defined using the Euclidean distance metric is also closely related to the Blum medial axis, a well known skeletal representation. Voronoi diagrams and medial axes have been shown useful for many 3D computations and operations, including proximity queries, motion planning, mesh generation, finite element analysis, and shape analysis. However, their application to complex 3D polyhedral and deformable models has been limited. This is due to the difficulty of computing exact Voronoi diagrams in an efficient and reliable manner. In this dissertation, we bridge this gap by presenting efficient algorithms to compute discrete Voronoi diagrams and simplified medial axes of 3D polyhedral models with geometric and topological guarantees. We apply these algorithms to complex 3D models and use them to perform interactive proximity queries, motion planning and skeletal computations. We present three new results. First, we describe an algorithm to compute 3D distance fields of geometric models by using a linear factorization of Euclidean distance vectors. This formulation maps directly to the linearly interpolating graphics rasterization hardware and enables us to compute distance fields of complex 3D models at interactive rates. We also use clamping and culling algorithms based on properties of Voronoi diagrams to accelerate this computation. We introduce surface distance maps, which are a compact distance vector field representation based on a mesh parameterization of triangulated two-manifolds, and use them to perform proximity computations. Our second main result is an adaptive sampling algorithm to compute an approximate Voronoi diagram that is homotopy equivalent to the exact Voronoi diagram and preserves topological features. We use this algorithm to compute a homotopy-preserving simplified medial axis of complex 3D models. Our third result is a unified approach to perform different proximity queries among multiple deformable models using second order discrete Voronoi diagrams. We introduce a new query called N-body distance query and show that different proximity queries, including collision detection, separation distance and penetration depth can be performed based on Nbody distance query. We compute the second order discrete Voronoi diagram using graphics hardware and use distance bounds to overcome the sampling errors and perform conservative computations. We have applied these queries to various deformable simulations and observed up to an order of magnitude improvement over prior algorithms

Efficient configuration space construction and optimization

The configuration space is a fundamental concept that is widely used in algorithmic robotics. Many applications in robotics, computer-aided design, and related areas can be reduced to computational problems in terms of configuration spaces. In this dissertation, we address three main computational challenges related to configuration spaces: 1) how to efficiently compute an approximate representation of high-dimensional configuration spaces; 2) how to efficiently perform geometric, proximity, and motion planning queries in high dimensional configuration spaces; and 3) how to model uncertainty in configuration spaces represented by noisy sensor data. We present new configuration space construction algorithms based on machine learning and geometric approximation techniques. These algorithms perform collision queries on many configuration samples. The collision query results are used to compute an approximate representation for the configuration space, which quickly converges to the exact configuration space. We highlight the efficiency of our algorithms for penetration depth computation and instance-based motion planning. We also present parallel GPU-based algorithms to accelerate the performance of optimization and search computations in configuration spaces. In particular, we design efficient GPU-based parallel k-nearest neighbor and parallel collision detection algorithms and use these algorithms to accelerate motion planning. In order to extend configuration space algorithms to handle noisy sensor data arising from real-world robotics applications, we model the uncertainty in the configuration space by formulating the collision probabilities for noisy data. We use these algorithms to perform reliable motion planning for the PR2 robot.Doctor of Philosoph

A fast method for local penetration depth computation

This paper presents a fast method for determining an approximation of the local penetration information for intersecting polyhedral models. As opposed to most techniques, this algorithm requires no specific knowledge of the object’s geometry or topology, or any pre-processing computations. In order to achieve real-time performance even for complex, non-convex models, we decouple the computation of the local penetration directions from the computation of the corresponding local penetration depths: for any pair of intersecting objects, we partition the penetrating zones into coherent regions, and we determine for each of these regions a local penetration direction. Then, for each of these regions, we estimate a local penetration depth in the previously computed penetration direction. This method has been implemented and tested on a 2.0 GHz Pentium PC with a NVIDIA GeForce FX 5900 graphics card with AGP 4 × and 768MB of RAM. The results indicate that a meaningful local penetration information can be computed for complex models and challenging intersection scenarios within a few milliseconds.