373 research outputs found
Trifocal Relative Pose from Lines at Points and its Efficient Solution
We present a new minimal problem for relative pose estimation mixing point
features with lines incident at points observed in three views and its
efficient homotopy continuation solver. We demonstrate the generality of the
approach by analyzing and solving an additional problem with mixed point and
line correspondences in three views. The minimal problems include
correspondences of (i) three points and one line and (ii) three points and two
lines through two of the points which is reported and analyzed here for the
first time. These are difficult to solve, as they have 216 and - as shown here
- 312 solutions, but cover important practical situations when line and point
features appear together, e.g., in urban scenes or when observing curves. We
demonstrate that even such difficult problems can be solved robustly using a
suitable homotopy continuation technique and we provide an implementation
optimized for minimal problems that can be integrated into engineering
applications. Our simulated and real experiments demonstrate our solvers in the
camera geometry computation task in structure from motion. We show that new
solvers allow for reconstructing challenging scenes where the standard two-view
initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1439786 while most authors were in residence
at Brown University's Institute for Computational and Experimental Research
in Mathematics -- ICERM, in Providence, R
ADE Spectral Networks
We introduce a new perspective and a generalization of spectral networks for
4d theories of class associated to Lie algebras
, , , and
. Spectral networks directly compute the BPS spectra of 2d
theories on surface defects coupled to the 4d theories. A Lie algebraic
interpretation of these spectra emerges naturally from our construction,
leading to a new description of 2d-4d wall-crossing phenomena. Our construction
also provides an efficient framework for the study of BPS spectra of the 4d
theories. In addition, we consider novel types of surface defects associated
with minuscule representations of .Comment: 68 pages plus appendices; visit
http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that
generates spectral networks; v2: version published in JHEP plus minor
correction
Incremental Sampling-based Algorithms for Optimal Motion Planning
During the last decade, incremental sampling-based motion planning
algorithms, such as the Rapidly-exploring Random Trees (RRTs) have been shown
to work well in practice and to possess theoretical guarantees such as
probabilistic completeness. However, no theoretical bounds on the quality of
the solution obtained by these algorithms have been established so far. The
first contribution of this paper is a negative result: it is proven that, under
mild technical conditions, the cost of the best path in the RRT converges
almost surely to a non-optimal value. Second, a new algorithm is considered,
called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost
of the best path in the RRG converges to the optimum almost surely. Third, a
tree version of RRG is introduced, called the RRT algorithm, which
preserves the asymptotic optimality of RRG while maintaining a tree structure
like RRT. The analysis of the new algorithms hinges on novel connections
between sampling-based motion planning algorithms and the theory of random
geometric graphs. In terms of computational complexity, it is shown that the
number of simple operations required by both the RRG and RRT algorithms is
asymptotically within a constant factor of that required by RRT.Comment: 20 pages, 10 figures, this manuscript is submitted to the
International Journal of Robotics Research, a short version is to appear at
the 2010 Robotics: Science and Systems Conference
Displacement Analysis of Under-Constrained Cable-Driven Parallel Robots
This dissertation studies the geometric static problem of under-constrained cable-driven
parallel robots (CDPRs) supported by n cables, with n ≤ 6. The task consists of determining the overall robot configuration when a set of n variables is assigned. When variables
relating to the platform posture are assigned, an inverse geometric static problem (IGP)
must be solved; whereas, when cable lengths are given, a direct geometric static problem (DGP) must be considered. Both problems are challenging, as the robot continues to
preserve some degrees of freedom even after n variables are assigned, with the final configuration determined by the applied forces. Hence, kinematics and statics are coupled and
must be resolved simultaneously.
In this dissertation, a general methodology is presented for modelling the aforementioned
scenario with a set of algebraic equations. An elimination procedure is provided, aimed at
solving the governing equations analytically and obtaining a least-degree univariate polynomial in the corresponding ideal for any value of n. Although an analytical procedure
based on elimination is important from a mathematical point of view, providing an upper
bound on the number of solutions in the complex field, it is not practical to compute these
solutions as it would be very time-consuming. Thus, for the efficient computation of the
solution set, a numerical procedure based on homotopy continuation is implemented. A
continuation algorithm is also applied to find a set of robot parameters with the maximum
number of real assembly modes for a given DGP. Finally, the end-effector pose depends
on the applied load and may change due to external disturbances. An investigation into
equilibrium stability is therefore performed
Asymptotics of multivariate sequences, part III: quadratic points
We consider a number of combinatorial problems in which rational generating
functions may be obtained, whose denominators have factors with certain
singularities. Specifically, there exist points near which one of the factors
is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the
coefficients of such a generating function. The computation requires some
topological deformations as well as Fourier-Laplace transforms of generalized
functions. We apply the results of the theory to specific combinatorial
problems, such as Aztec diamond tilings, cube groves, and multi-set
permutations.Comment: substantial correction
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