Peter F. Stiller

MAPRM: a probabilistic roadmap planner with sampling on the medial axis of 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.

Dexterous manipulation planning and execution of an enveloped slippery workpiece

When robotic hands or arms are capable of enveloping the workpieces that they manipulate, envelopment of the workpiece ensures grasp maintenance even if the object experiences significant external forces in directions unknown prior to grasp synthesis. The enveloping mechanism is useful in low-friction and microgravity environments. First-order stability cells are defined. They are used to plan a planar, whole-arm, manipulation task of a slippery workpiece. For most, but not all, of the plan, the workpiece is enveloped. Experimental results are presented.<<ETX>>

Motion planning for a rigid body using random networks on the medial axis of the free space

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 free space. In previous work we proposed MAPRM, a method of sampling the configuration space in which randomly generated 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 rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the C-obstacles or the medial axis. We give theoretical and experimental results to show this improves performance on problems involving narrow corridors and compare the performance to uniform random sampling from the free space.

Special values of Dirichlet series, monodromy, and the periods of automorphic forms

Introduction Periods of automorphic forms Locally split extensions of flat vector bundles bis. Automorphic forms attached to the symmetric powers of a second order differential equation and their periods Descent Differential equations associated to automorphic forms Examples.

Solving the recognition problem for six lines using the Dixon resultant

The “Six-line Problem” arises in computer vision and in the automated analysis of images. Given a three-dimensional (3D) object, one extracts geometric features (for example six lines) and then, via techniques from algebraic geometry and geometric invariant theory, produces a set of 3D invariants that represents that feature set. Suppose that later an object is encountered in an image (for example, a photograph taken by a camera modeled by standard perspective projection, i.e. a “pinhole” camera), and suppose further that six lines are extracted from the object appearing in the image. The problem is to decide if the object in the image is the original 3D object. To answer this question two-dimensional (2D) invariants are computed from the lines in the image. One can show that conditions for geometric consistency between the 3D object features and the 2D image features can be expressed as a set of polynomial equations in the combined set of two- and three-dimensional invariants. The object in the image is geometrically consistent with the original object if the set of equations has a solution. One well known method to attack such sets of equations is with resultants. Unfortunately, the size and complexity of this problem made it appear overwhelming until recently. This paper will describe a solution obtained using our own variant of the Cayley–Dixon–Kapur–Saxena–Yang resultant. There is reason to believe that the resultant technique we employ here may solve other complex polynomial systems.

Second-order stability cells of a frictionless rigid body grasped by rigid fingers

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 systems 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.<<ETX>>

Classical automorphic forms and hypergeometric functions

We exhibit a graded algebra of hypergeometric functions and show how to canonically identify it with the graded algebra of modular forms for the full modular group SL2(Z). We also show how Dedekinds eta function is related to the square root of a hypergeometric function and give yet another simple proof of its functional equation. The methods permit the simple translation of integrals of modular forms (e.g., Mellin transforms and special values of associated Dirichlet series) into integrals of hypergeometric functions where the theory of these classical special functions can be brought to bear. As an example, we express the Mellin transform of an Eisenstein series (which involves Riemanns zeta function) in terms of hypergeometric functions.

On the geometry of contact formation cells for systems of polygons

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 systems 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. >

On the algebraic geometry of contact formation cells for systems of polygons

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.<<ETX>>

Certain reflexive sheaves on ⁿ_{} and a problem in approximation theory

This paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces.