272 research outputs found
Reconstruction of Support of a Measure From Its Moments
In this paper, we address the problem of reconstruction of support of a
measure from its moments. More precisely, given a finite subset of the moments
of a measure, we develop a semidefinite program for approximating the support
of measure using level sets of polynomials. To solve this problem, a sequence
of convex relaxations is provided, whose optimal solution is shown to converge
to the support of measure of interest. Moreover, the provided approach is
modified to improve the results for uniform measures. Numerical examples are
presented to illustrate the performance of the proposed approach.Comment: This has been submitted to the 53rd IEEE Conference on Decision and
Contro
An identity theorem for the Fourier transform of polytopes on rationally parameterisable hypersurfaces
A set of points in is called a rationally
parameterisable hypersurface if , where is a vector function with domain and rational functions as
components. A generalized -dimensional polytope in is a union
of a finite number of convex -dimensional polytopes in . The
Fourier transform of such a generalized polytope in
is defined by . We prove that
implies if is an open subset of
satisfying some well-defined conditions. Moreover we show that this theorem can
be applied to quadric hypersurfaces that do not contain a line, but at least
two points, i.e., in particular to spheres.Comment: 20 page
Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms
We propose strongly consistent algorithms for reconstructing the
characteristic function 1_K of an unknown convex body K in R^n from possibly
noisy measurements of the modulus of its Fourier transform \hat{1_K}. This
represents a complete theoretical solution to the Phase Retrieval Problem for
characteristic functions of convex bodies. The approach is via the closely
related problem of reconstructing K from noisy measurements of its covariogram,
the function giving the volume of the intersection of K with its translates. In
the many known situations in which the covariogram determines a convex body, up
to reflection in the origin and when the position of the body is fixed, our
algorithms use O(k^n) noisy covariogram measurements to construct a convex
polytope P_k that approximates K or its reflection -K in the origin. (By recent
uniqueness results, this applies to all planar convex bodies, all
three-dimensional convex polytopes, and all symmetric and most (in the sense of
Baire category) arbitrary convex bodies in all dimensions.) Two methods are
provided, and both are shown to be strongly consistent, in the sense that,
almost surely, the minimum of the Hausdorff distance between P_k and K or -K
tends to zero as k tends to infinity.Comment: Version accepted on the Journal of the American Mathematical Society.
With respect to version 1 the noise model has been greatly extended and an
appendix has been added, with a discussion of rates of convergence and
implementation issues. 56 pages, 4 figure
Actuation-Aware Simplified Dynamic Models for Robotic Legged Locomotion
In recent years, we witnessed an ever increasing number of successful hardware implementations of motion planners for legged robots. If one common property is to be identified among these real-world applications, that is the ability of online planning.
Online planning is forgiving, in the sense that it allows to relentlessly compensate for external disturbances of whatever form they might be, ranging from unmodeled dynamics to external pushes or unexpected obstacles and, at the same time, follow user commands. Initially replanning was restricted only to heuristic-based planners that exploit the low computational effort of simplified dynamic models. Such models deliberately only capture the main dynamics of the system, thus leaving to the controllers the issue of anchoring the desired trajectory to the whole body model of the robot. In recent years, however, we have seen a number of new approaches attempting to increase the accuracy of the dynamic formulation without trading-off the computational efficiency of simplified models.
In this dissertation, as an example of successful hardware implementation of heuristics and simplified model-based locomotion, I describe the framework that I developed for the generation of an omni-directional bounding gait for the HyQ quadruped robot. By analyzing the stable limit cycles for the sagittal dynamics and the Center of Pressure (CoP) for the lateral stabilization, the described locomotion framework is able to achieve a stable bounding while adapting to terrains of mild roughness and to sudden changes of the user desired linear and angular velocities.
The next topic reported and second contribution of this dissertation is my effort to formulate more descriptive simplified dynamic models, without trading off their computational efficiency, in order to extend the navigation capabilities of legged robots to complex geometry environments. With this in mind, I investigated the possibility of incorporating feasibility constraints in these template models and, in particular, I focused on the joint torques limits which are usually neglected at the planning stage.
In this direction, the third contribution discussed in this thesis is the formulation of the so called actuation wrench polytope (AWP), defined as the set of feasible wrenches that an articulated robot can perform given its actuation limits. Interesected with the contact wrench cone (CWC), this yields a new 6D polytope that we name feasible wrench polytope (FWP), defined as the set of all wrenches that a legged robot can realize given its actuation capabilities and the friction constraints. Results are reported where, thanks to efficient computational geometry algorithms and to appropriate approximations, the FWP is employed for a one-step receding horizon optimization of center of mass trajectory and phase durations given a predefined step sequence on rough terrains.
For the sake of reachable workspace augmentation, I then decided to trade off the generality of the FWP formulation for a suboptimal scenario in which a quasi-static motion is assumed.
This led to the definition of the, so called, local/instantaneous actuation region and of the global actuation/feasible region. They both can be seen as different variants of 2D linear subspaces orthogonal to gravity where the robot is guaranteed to place its own center of mass while being able to carry its own body weight given its actuation capabilities. These areas can be intersected with the well known frictional support region, resulting in a 2D linear feasible region, thus providing an intuitive tool that enables the concurrent online optimization of actuation consistent CoM trajectories and target foothold locations on rough terrains
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
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