A Motion Planning Processor on Reconfigurable Hardware

Motion planning algorithms enable us to find feasible paths for moving objects. These algorithms utilize feasibility checks to differentiate valid paths from invalid ones. Unfortunately, the computationally expensive nature of such checks reduces the effectiveness of motion planning algorithms. However, by using hardware acceleration to speed up the feasibility checks, we can greatly enhance the performance of the motion planning algorithms. Of course, such acceleration is not limited to feasibility checks; other components of motion planning algorithms can also be accelerated using specially designed hardware. A Field Programmable Gate Array (FPGA) is a great platform to support such an acceleration. An FPGA is a collection of digital gates which can be reprogrammed at run time, i.e., it can be used as a CPU that reconfigures itself for a given task. In this paper, we study the feasibility of an FPGA based motion planning processor and evaluate its performance. In order to leverage its highly parallel nature and its modular structure, our processor utilizes the probabilistic roadmap method at its core. The modularity enables us to replace the feasibility criteria with other ones. The reconfigurability lets us run our processor in different roles, such as a motion planning co-processor, an autonomous motion planning processor or dedicated collision detection chip. Our experiments show that such a processor is not only feasible but also can greatly increase the performance of current algorithms

Minimizing the error of linear separators on linearly inseparable data

Given linearly inseparable sets R of red points and B of blue points, we consider several measures of how far they are from being separable. Intuitively, given a potential separator (‘‘classifier’’), we measure its quality (‘‘error’’) according to how much work it would take to move the misclassified points across the classifier to yield separated sets. We consider several measures of work and provide algorithms to find linear classifiers that minimize the error under these different measures.Ministerio de Educación y Ciencia MTM2008-05866-C03-0

Interactive Hausdorff distance computation for general polygonal models

Figure 1: Interactive Hausdorff Distance Computation. Our algorithm can compute Hausdorff distance between complicated models at interactive rates (the first three figures). Here, the green line denotes the Hausdorff distance. This algorithm can also be used to find penetration depth (PD) for physically-based animation (the last two figures). It takes only a few milli-seconds to run on average. We present a simple algorithm to compute the Hausdorff distance between complicated, polygonal models at interactive rates. The algorithm requires no assumptions about the underlying topology and geometry. To avoid the high computational and implementa-tion complexity of exact Hausdorff distance calculation, we approx-imate the Hausdorff distance within a user-specified error bound. The main ingredient of our approximation algorithm is a novel polygon subdivision scheme, called Voronoi subdivision, combined with culling between the models based on bounding volume hier-archy (BVH). This cross-culling method relies on tight yet simple computation of bounds on the Hausdorff distance, and it discards unnecessary polygon pairs from each of the input models alterna-tively based on the distance bounds. This algorithm can approxi-mate the Hausdorff distance between polygonal models consisting of tens of thousands triangles with a small error bound in real-time, and outperforms the existing algorithm by more than an order of magnitude. We apply our Hausdorff distance algorithm to the mea-surement of shape similarity, and the computation of penetration depth for physically-based animation. In particular, the penetration depth computation using Hausdorff distance runs at highly interac-tive rates for complicated dynamics scene

A constraint-stabilized time-stepping approach for piecewise smooth multibody dynamics

Rigid multibody dynamics is an important area of mathematical modeling which attempts to predict the position and velocity of a system of rigid bodies. Many methods will use smooth bodies without friction. The task is made especially more difficult in the face of noninterpenetration constraints, joint constraints, and friction forces. The difficulty that arises when noninterpenetration constraints are enforced is directly related to the fact that the usual methods of computing the distance between bodies do not give any indication of the amount of penetration when two bodies interpenetrate. Because we wish to calculate vectors that are normal to contact, and because it is necessary to determine the amount of penetration, when it exists, the classical computation of the depth of penetration when applied to convex polyhedral bodies is inefficient.We hereby describe a new method of determining when two convex polyhedra intersect and of evaluating a measure of the amount of penetration, when it exists. Our method is much more efficient than the classic computation of the penetration depth since it can be shown that its complexity grows only linearly with the size of the problem. We use our method to construct a signed distance function and implement it for use with a method for achieving geometrical constraint stabilization for a linear-complementarity-based time-stepping scheme for rigid multibody dynamics with joints, contact, and friction which, before now, was not equipped to handle polyhedral bodies. During our analysis, we describe how to compute normal vectors at contact, despite the cases when the classic derivative fails to exist.We put this analysis into a time-stepping procedure that uses a convex relaxation of a mixed linear complementarity problem with a resulting fixed point iteration that is guaranteed to converge if the friction is not too large, the time step is not too large, and the initial solution is feasible. Finally, we construct an algorithm that achieves constraint stabilization with quadratic convergence.The numerical results proved to be quite satisfactory, implying that the constraint stabilization holds, and that quadratic convergence exists

Fast Penetration Depth Computation for Physically-based Animation

We present a novel and fast algorithm to compute penetration depth (PD) between two polyhedral models for physically-based animation. Given two overlapping polyhedra, it computes the minimal translation distance to separate them using a combination of object-space and image-space techniques. The algorithm computes pairwise Minkowski sums of decomposed convex pieces and performs a closest point query using rasterization hardware. It uses bounding volume hierarchies, object-space and image-space culling algorithms to further accelerate the computation and refines the estimated PD in a hierarchical manner. We demonstrate its application to contact response computation and a time-stepping method for dynamic simulation

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