Dynamic Detection of Topological Information from Grid-Based Generalized Voronoi Diagrams

In the context of robotics, the grid-based Generalized Voronoi Diagrams (GVDs) are widely used by mobile robots to represent their surrounding area. Current approaches for incrementally constructing GVDs mainly focus on providing metric skeletons of underlying grids, while the connectivity among GVD vertices and edges remains implicit, which makes high-level spatial reasoning tasks impractical. In this paper, we present an algorithm named Dynamic Topology Detector (DTD) for extracting a GVD with topological information from a grid map. Beyond the construction and reconstruction of a GVD on grids, DTD further extracts connectivity among the GVD edges and vertices. DTD also provides efficient repair mechanism to treat with local changes, making it work well in dynamic environments. Simulation tests in representative scenarios demonstrate that (1) compared with the static algorithms, DTD generally makes an order of magnitude improvement regarding computation times when working in dynamic environments; (2) with negligible extra computation, DTD detects topologies not computed by existing incremental algorithms. We also demonstrate the usefulness of the resulting topological information for high-level path planning tasks

Real-time reach planning for animated characters using hardware acceleration

We present a heuristic-based real-time reach planning algorithm for virtual human figures. Given the start and goal positions in a 3D workspace, our problem is to compute a collision-free path that specifies all the configurations for a human arm to move from the start to the goal. Our algorithm consists of three modules: spatial search, inverse kinematics, and collision detection. For the search module, instead of searching in joint configuration space like most existing motion planning methods do, we run a direct search in the workspace, guided by a heuristic distance-to-goal evaluation function. The inverse kinematics module attempts to select natural posture configurations for the arm along the path found in the workspace. During the search, candidate configurations will be checked for collisions taking advantage of the graphics hardware – depth buffer. The algorithm is fast and easy to implement. It allows real-time planning not only in static, structured environments, but also in dynamic, unstructured environments. No preprocessing and prior knowledge about the environment is required. Several examples are shown illustrating the competence of the planner at generating motion plans for a typical human arm model with seven degrees of freedom

Exact Generalized Voronoi Diagram Computation using a Sweepline Algorithm

Voronoi Diagrams can provide useful spatial information. Little work has been done on computing exact Voronoi Diagrams when the sites are more complex than a point. We introduce a technique that measures the exact Generalized Voronoi Diagram from points, line segments and, connected lines including lines that connect to form simple polygons. Our technique is an extension of Fortune’s method. Our approach treats connected lines (or polygons) as a single site

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

Path planning for complex 3D multilevel environments

The continuous development of graphics hardware is contributing to the creation of 3D virtual worlds with high level of detail, from models of large urban areas, to complete infrastructures, such as residential buildings, stadiums, industrial settings or archaeological sites, to name just a few. Adding virtual humans or avatars adds an extra touch to the visualization providing an enhanced perception of the spaces, namely adding a sense of scale, and enabling simulations of crowds. Path planning for crowds in a meaningful way is still an open research field, particularly when it involves an unknown polygonal 3D world. Extracting the potential paths for navigation in a non automated fashion is no longer a feasible option due to the dimension and complexity of the virtual environments available nowadays. This implies that we must be able to automatically extract information from the geometry of the unknown virtual world to define potential paths, determine accessibilities, and prepare a navigation structure for real time path planning and path finding. A new image based method is proposed that deals with arbitrarily a priori unknown complex virtual worlds, namely those consisting of multilevel passages (e.g. over and below a bridge). The algorithm is capable of extracting all the information required for the actual navigation of avatars, creating a hierarchical data structure to help both high level path planning and low level path finding decisions. The algorithm is image based, hence it is tessellation independent, i.e. the algorithm does not rely on the underlying polygonal structure of the 3D world. Therefore, the number of polygons does not have a significant impact on the performance, and the topology has no weight on the results.Fundação para a Ciência e a Tecnologi

Path planning for complex 3D multilevel environments

The continuous development of graphics hardware is contributing to the creation of 3D virtual worlds with high level of detail, from models of large urban areas, to complete infrastructures, such as residential buildings, stadiums, industrial settings or archaeological sites, to name just a few. Adding virtual humans or avatars adds an extra touch to the visualization providing an enhanced perception of the spaces, namely adding a sense of scale, and enabling simulations of crowds. Path planning for crowds in a meaningful way is still an open research field, particularly when it involves an unknown polygonal 3D world. Extracting the potential paths for navigation in a non automated fashion is no longer a feasible option due to the dimension and complexity of the virtual environments available nowadays. This implies that we must be able to automatically extract information from the geometry of the unknown virtual world to define potential paths, determine accessibilities, and prepare a navigation structure for real time path planning and path finding. A new image based method is proposed that deals with arbitrarily a priori unknown complex virtual worlds, namely those consisting of multilevel passages (e.g. over and below a bridge). The algorithm is capable of extracting all the information required for the actual navigation of avatars, creating a hierarchical data structure to help both high level path planning and low level path finding decisions. The algorithm is image based, hence it is tessellation independent, i.e. the algorithm does not use the underlying polygonal structure of the 3D world. Therefore, the number of polygons as well as the topology, do not affect the performance