Federico Thomas

3D collision detection: a survey

Abstract Many applications in Computer Graphics require fast and robust 3D collision detection algorithms. These algorithms can be grouped into four approaches: space–time volume intersection, swept volume interference, multiple interference detection and trajectory parameterization. While some approaches are linked to a particular object representation scheme (e.g., space–time volume intersection is particularly suited to a CSG representation), others do not. The multiple interference detection approach has been the most widely used under a variety of sampling strategies, reducing the collision detection problem to multiple calls to static interference tests. In most cases, these tests boil down to detecting intersections between simple geometric entities, such as spheres, boxes aligned with the coordinate axes, or polygons and segments. The computational cost of a collision detection algorithm depends not only on the complexity of the basic interference test used, but also on the number of times this test is applied. Therefore, it is crucial to apply this test only at those instants and places where a collision can truly occur. Several strategies have been developed to this end: (1) to find a lower time bound for the first collision, (2) to reduce the pairs of primitives within objects susceptible of interfering, and (3) to cut down the number of object pairs to be considered for interference. These strategies rely on distance computation algorithms, hierarchical object representations, orientation-based pruning criteria, and space partitioning schemes. This paper tries to provide a comprehensive survey of all these techniques from a unified viewpoint, so that well-known algorithms are presented as particular instances of general approaches.

Revisiting trilateration for robot localization

Locating a robot from its distances, or range measurements, to three other known points or stations is a common operation, known as trilateration. This problem has been traditionally solved either by algebraic or numerical methods. An approach that avoids the direct algebrization of the problem is proposed here. Using constructive geometric arguments, a coordinate-free formula containing a small number of Cayley-Menger determinants is derived. This formulation accommodates a more thorough investigation of the effects caused by all possible sources of error, including round-off errors, for the first time in this context. New formulas for the variance and bias of the unknown robot location estimation, due to station location and range measurements errors, are derived and analyzed. They are proved to be more tractable compared with previous ones, because all their terms have geometric meaning, allowing a simple analysis of their asymptotic behavior near singularities.

An ellipsoidal calculus based on propagation and fusion

Presents an ellipsoidal calculus based solely on two basic operations: propagation and fusion. Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the intersection of two given ellipsoids. These two operations supersede the Minkowski sum and difference, affine transformation and intersection tight bounding of ellipsoids on which other ellipsoidal calculi are based. Actually, a Minkowski operation can be seen as a fusion followed by a propagation and an affine transformation as a particular case of propagation. Moreover, the presented formulation is numerically stable in the sense that it is immune to degeneracies of the involved ellipsoids and/or affine relations. Examples arising when manipulating uncertain geometric information in the context of the spatial interpretation of line drawings are extensively used as a testbed for the presented calculus.

A group-theoretic approach to the computation of symbolic part relations

When a set of constraints is imposed on the degrees of freedom between several rigid bodies, finding the configuration or configurations that satisfy all these constraints is a matter of special interest. The problem is not new and has been discussed, not only in kinematics, but also more recently in the design of object-level robot programming languages. In this last domain, several languages have been developed, from different points of view, that are able to partially solve the problem. A more general method is derived than those previously proposed that were based on the symbolic manipulation of chains of matrix products, using the theory of continuous groups. >

Set membership approach to the propagation of uncertain geometric information

An alternative approach for the propagation of uncertain geometric information, based on the ideas presented by J.R. Deller (IEEE ASSP Magazine, vol.6, p.4-20, Oct. 1989) and extended to deal with graphs of geometric constraints, is presented. This method avoids the independency assumption of the probabilistic approach. In this approach, when new sensory data are acquired, a set of strips is obtained, propagated, and fused to obtain the updated ellipsoids associated with each feature, Then, the hypothesis about the location of the involved geometric features can be easily updated. Inconsistencies are easily detected, resulting in fast rejection of erroneous data.<<ETX>>

A Linear Relaxation Technique for the Position Analysis of Multiloop Linkages

This paper presents a new method to isolate all configurations that a multiloop linkage can adopt. The problem is tackled by means of formulation and resolution techniques that fit particularly well together. The adopted formulation yields a system of simple equations (only containing linear, bilinear, and quadratic monomials, and trivial trigonometric terms for the helical pair only) whose structure is later exploited by a branch-and-prune method based on linear relaxations. The method is general, as it can be applied to linkages with single or multiple loops with arbitrary topology, involving lower pairs of any kind, and complete, as all possible solutions get accurately bounded, irrespective of whether the linkage is rigid or mobile.

Efficient computation of local geometric moments

Local moments have attracted attention as local features in applications such as edge detection and texture segmentation. The main reason for this is that they are inherently integral-based features, so that their use reduces the effect of uncorrelated noise. The computation of local moments, when viewed as a neighborhood operation, can be interpreted as a convolution of the image with a set of masks. Nevertheless, moments computed inside overlapping windows are not independent and convolution does not take this fact into account. By introducing a matrix formulation and the concept of accumulation moments, this paper presents an algorithm which is computationally much more efficient than convolving and yet as simple.

Interference detection between non-convex polyhedra revisited with a practical aim

Exact interference checking between two arbitrary polyhedra is known to have O(mn) complexity, where m and n are the number of edges in the two polyhedra. This is just a worst-case bound that still leaves plenty of room for algorithm improvement in practice. The algorithm presented herein has been developed so as to: 1. Minimize the number of operations that each pairing of edges entails. We prove that this factor is 4.5 for multiplications and 8.5 for additions. 2. Avoid the construction of auxiliary geometric entities. The standard approach is to decompose the nonconvex polyhedra (or their faces) into convex entities and then check for interference in this convex setting. This entails the construction of many fictitious edges and faces, which indirectly contribute to the growth of the complexity. 3. Permit the straightforward application of prunning strategies to most practical situations, so that the worst-case bound above is reached only when truly needed. 4 Allow the derivation of both directional and undirectional distance bounds between the polyhedra, which prove extremely useful for collision avoidance and local path planning. The simplicity and homogeneity of the algorithm has led to a quick implementation, which has been proven to be robust and fast. Some performance measurements are reported.<<ETX>>

Complete maps of molecular‐loop conformational spaces

This paper presents a numerical method to compute all possible conformations of distance‐constrained molecular loops, i.e., loops where some interatomic distances are held fixed, while others can vary. The method is general (it can be applied to single or multiple intermingled loops of arbitrary topology) and complete (it isolates all solutions, even if they form positive‐dimensional sets). Generality is achieved by reducing the problem to finding all embeddings of a set of points constrained by pairwise distances, which can be formulated as computing the roots of a system of Cayley–Menger determinants. Completeness is achieved by expressing these determinants in Bernstein form and using a numerical algorithm that exploits such form to bound all root locations at any desired precision. The method is readily parallelizable, and the current implementation can be run on single‐ or multiprocessor machines. Experiments are included that show the methods performance on rigid loops, mobile loops, and multiloop molecules. In all cases, complete maps including all possible conformations are obtained, thus allowing an exhaustive analysis and visualization of all pseudo‐rotation paths between different conformations satisfying loop closure.

A branch-and-prune solver for distance constraints

Given some geometric elements such as points and lines in R/sup 3/, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branch-and-prune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one.