In this paper we survey some of the mathematical techniques that have led to useful new results in shape analysis and their application to a variety of object recognition tasks. In particular, we will show how these techniques allow one to solve a number of fundamental problems related to object recognition for configurations of point features under a generalized weak perspective model of image formation. Our approach makes use of progress in shape theory and includes the development of object-image equations for shape matching and the exploitation of shape space metrices (especially object-image metrics) to measure matching up to certain transformations. This theory is built on advanced mathematical techniques from algebraic and differential geometry which are used to construct generalized shape spaces for various projection and sensor models. That construction in turn is used to find natural metrics that express the distance (geometric difference) between two configurations of object features, two configurations of image features, or an object and an image pair. Such metrics are believed to produce the most robust tests for object identification; at least as far as the object's geometry is concerned. Moreover, these metrics provide a basis for efficient hashing schemes to do identification quickly, and they provide a rigorous foundation for error and statistical analysis in any recognition system. The most important feature of a shape theoretic approach is that all of the matching tests and metrics are independent of the choice of coordinates used to express the feature locations on the object or in the image. In addition, the approach is independent of the camera/sensor position and any camera/sensor parameters. Finally, the method is also independent of object pose or image orientation. This is what makes the results so powerful.

Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required to pass through narrow passages in the,jree space.In [l S] we proposed MAPRM, a method of sampling the configumtion space in which randomly genemted configurations, free or not, are retracted onto the medial axis of the free space without having to first compute the medial axis; this was shown to increase sampling in narrow passages.In this paper we give details of the MAPRM algorithm for the case of a free-flying rigid body moving in three dimensions, and show that the retmction may be carried out without explicitly computing the Cobstacles or the medial axis.We give theoretical arguments to show that this improves sampling in narrow corridors, and present preliminary experimental results comparing the performance to uniform random sampling from the free space.

Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a new method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. We give algorithms that perform this retraction while avoiding explicit computation of the medial axis, and we show that sampling and retracting in this manner increases the number of nodes found in small volume corridors in a way that is independent of the volume of the corridor and depends only on the characteristics of the obstacles bounding it. Theoretical and experimental results are given to show that this improves performance on problems requiring traversal of narrow passages.

Article Free Access Share on Symbolic computation of object/image equations Author: Peter F. Stiller Department of Mathematics and Department of Computer Science, Texas A&M University Department of Mathematics and Department of Computer Science, Texas A&M UniversityView Profile Authors Info & Claims ISSAC '97: Proceedings of the 1997 international symposium on Symbolic and algebraic computationJuly 1997 Pages 359–364https://doi.org/10.1145/258726.258846Published:01 July 1997Publication History 4citation166DownloadsMetricsTotal Citations4Total Downloads166Last 12 Months12Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The contact formation cells of a polygonal planar system of rigid bodies in contact have been studied in Farahat et al. There, it was shown that the CF-cells are smooth manifolds, but the methods used were too complicated to extend to three-dimensional polygonal rigid body systems. In this paper, the authors develop an alternative way to define contact formation cells. Under the new definition, the authors show that the contact formation cells are smooth manifolds, and further that all intersections of contact formation cells are smooth manifolds. The simplicity of the new definition makes it easy to prove the smoothness results for three-dimensional systems. Also, the authors investigate other extensions of the results in Farahat et al.

The efficient planning of contact tasks for intelligent robotic systems requires a thorough understanding of the kinematic constraints imposed on the system by the contacts. In this paper, we derive closed-form analytic solutions for the position and orientation of a passive polygon moving in sliding and rolling contact with two or three active polygons whose positions and orientations are independently controlled. This is accomplished by applying a simple elimination procedure to solve the appropriate system of contact constraint equations. We also prove that the set of solutions to the contact constraint equations is a smooth submanifold of the system's configuration space and we study its projection onto the configuration space of the active polygons. By relating these results to the wrench matrices commonly used in grasp analysis, we discover a previously unknown and highly nonintuitive class of nongeneric contact situations. In these situations, for a specific fixed configuration of the active polygons, the passive polygon can maintain three contacts on three mutually nonparallel edges while retaining one degree of freedom of motion.>

A stability cell is a subset of the configuration space (C-space) of a set of actively controlled rigid bodies (e.g., a manipulator) in contact with a passive body in which the contact state is guaranteed to be stable under Coulomb friction and external forces. A first-order stability cell is a subset of a stability cell with the following two properties: the state of contact uniquely determines the rate of change of the object's configuration given the rate of change of the manipulator's configuration; and the contact state cannot be altered by any infinitesimal variation in the generalized applied force. First-order stability cells can be used in planning whole-arm manipulation tasks in a manner analogous to the use of free-space cells in planning collision-free paths: a connectivity graph is constructed and searched for a path connecting the initial and goal configurations. A path through a free-space connectivity graph represents a motion plan that can be executed without fear of collisions, while a path through a stability-cell connectivity graph represents a whole-arm manipulation plan that can be executed without fear of "dropping" the object. The paper gives a conceptual and analytical development of first-order stability cells of 3D rigid-body systems as conjunctions of equations and inequalities in the C-space variables. Additionally, our derivation leads to a new quasi-static jamming condition that takes into account the planned motion and kinematic structure of the active bodies.>

The efficient planning of contact tasks for intelligent robotic systems requires a thorough understanding of the kinematic constraints imposed on the system by rolling and sliding contacts. In this paper, we derive closed-form analytic solutions for the position and orientation of a passive polygon moving in contact with two or three active polygons whose positions and orientations are independently controlled. This is done by applying elimination techniques to solve the systems of appropriate contact constraint equations. We prove that the systems of contact constraint equations are smooth submanifolds of configuration space.>

The most secure type of grasp of a frictionless workpiece is the form-closure grasp. However, task constraints may make achieving form-closure impossible or undesirable. In this case, one needs to employ a force-closure grasp. In this paper, we study the subclass of force-closure grasps known as second-order stable grasps, which typically have a small number of contacts. We derive conditions for second-order stability and represent second-order stability cells as conjunctions of equations and inequalities in the configuration variables of the system. These cells are the subsets of the system's configuration space for which the frictionless workpiece is second-order stable. We also determine the minimum and maximum numbers of contacts necessary for second-order stability. Our results are applied to a simple planar whole-arm manipulation system to generate one of its second-order stability cells.>