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We describe a method for determining affine and metric calibration of a camera with unchanging internal parameters undergoing planar motion. It is shown that affine calibration is recovered uniquely, and metric calibration up to a two fold ambiguity. The novel

We describe a method to determine affine and metric calibration for a stereo rig. The method does not involve the use of calibration objects or special motions, but simply a single general motion of the rig with fixed parameters (i.e. camera parameters and relative orientation of the camera pair). The novel aspects of this work are: first, relating the distinguished objects of Euclidean geometry to fixed entities of a Euclidean transformation matrix; second, showing that these fixed entities are accessible from the conjugate Euclidean transformation arising from the projective transformation of the structure under a motion of the fixed stereo rig; third, a robust and automatic implementation of the method. Results are included of affine and metric calibration and structure recovery using images of real scenes. 1 Introduction It is known that from two uncalibrated views, 3D structure can be recovered modulo a projectivity of 3space [5, 8]. Furthermore, given three or more views acquir...

This dissertation addresses the problem of generating feasible assembly sequences for a mechanical product from a geometric model of the product. An operation specifies a motion to bring two subassemblies together to make a larger subassembly. An assembly sequence is a sequence of operations that construct the product from the individual parts. I introduce the non-directional blocking graph, a succinct characterization of the blocking relationships between parts in an assembly. I describe efficient algorithms to identify removable subassemblies by constructing and analyzing the NDBG. For an assembly A of n parts and m part--part contacts equivalent to k contact points, a subassembly that can translate a small distance from the rest of A can be identified in O(mk 2 ) time. When rotations are allowed as well, the time bound is O(mk 5 ). Both algorithms are extended to find connected subassemblies in the same time bounds. All free subassemblies can be identified in output-dependent ...

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

: This paper addresses the problem of computing stable grasps of three-dimensional polyhedral objects. We consider the case of a hand equipped with four hard fingers and assume point contact with friction. We prove new necessary and sufficient conditions for equilibrium and force closure, and present a geometric characterization of all possible types of four-finger equilibrium grasps. We then focus on concurrent grasps, for which the lines of action of the four contact forces all intersect in a point. In this case, the equilibrium conditions are linear in the unknown grasp parameters, which reduces the problem of computing the stable grasp regions in configuration space to the problem of constructing the eight-dimensional projection of an eleven-dimensional polytope. We present two projection methods: the first one uses a simple Gaussian elimination approach, while the second one relies on a novel output-sensitive contour-tracking algorithm. Finally, we use linear optimization within t...

this paper we describe a method for achieving affine calibration for a fixed stereo rig on an AGV using constraints that arise naturally during motion - namely, that translation is in a plane perpendicular to the rotation axis. We describe and compare two approaches which employ these constraints: first, by recovering structure modulo a projectivity, and thence the plane at infinity; second, by using only image measurements based on the fundamental matrix (i.e. no structure recovery). The methods require no knowledge of the translation or angle of rotation. All processing - point correspondence based on corners, computation of the fundamental matrix, 3D structure recovery, affine calibration - is robust and automatic.

Abstract--General six-in-parallel SPS platform manipulators are constructed of six telescoping legs, each connecting a stationary base platform to a moving platform via spherical joints. These are often termed "generalized Stewart platforms. " Given the lengths of the six legs, the forward displacement problem is to find the location of the end platform relative to the base platform. It was first demonstrated numerically that the problem may in general have at most 40 nonsingular solutions and this bound has been verified using several different mathematical arguments. The problem is reformulated in this report using a classical representation of rigid-body displacements: Study's soma coordinates, or equivalently, dual quaternions. This provides a much simpler analytical proof of the upper bound of 40. Moreover, the simple form of the equations may be useful in further studies of the problem. 1.

Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.

In this work, we present a fundamentally new approach to visual servoing using lines. It is based on a theoretical and geometrical study of the main line representations which allowed us to define a new representation, the so-called binormalized Plücker coordinates. They are particularly well suited to visual servoing. Indeed, they allow the definition of a proper image line alignment notion. The control law which realizes such an alignment has moreover several properties: partial decoupling between rotation and translation, analytical inversion of the motion equations and global asymptotic stability conditions. This control law was validated both in simulation and experimentally in the specific case of an orthogonal trihedron.