357 research outputs found
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Convex Relaxations of SE(2) and SE(3) for Visual Pose Estimation
This paper proposes a new method for rigid body pose estimation based on
spectrahedral representations of the tautological orbitopes of and
. The approach can use dense point cloud data from stereo vision or an
RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data.
The method is a convex relaxation of the classical pose estimation problem, and
is based on explicit linear matrix inequality (LMI) representations for the
convex hulls of and . Given these representations, the relaxed
pose estimation problem can be framed as a robust least squares problem with
the optimization variable constrained to these convex sets. Although this
formulation is a relaxation of the original problem, numerical experiments
indicate that it is indeed exact - i.e. its solution is a member of or
- in many interesting settings. We additionally show that this method
is guaranteed to be exact for a large class of pose estimation problems.Comment: ICRA 2014 Preprin
Soft-connected Rigid Body Localization: State-of-the-Art and Research Directions for 6G
This white paper describes a proposed article that will aim to provide a
thorough study of the evolution of the typical paradigm of wireless
localization (WL), which is based on a single point model of each target,
towards wireless rigid body localization (W-RBL). We also look beyond the
concept of RBL itself, whereby each target is modeled as an independent
multi-point three-dimensional (3D), with shape enforced via a set of
conformation constraints, as a step towards a more general approach we refer to
as soft-connected RBL, whereby an ensemble of several objects embedded in a
given environment, is modeled as a set of soft-connected 3D objects, with rigid
and soft conformation constraints enforced within each object and among them,
respectively. A first intended contribution of the full version of this article
is a compact but comprehensive survey on mechanisms to evolve WL algorithms in
W-RBL schemes, considering their peculiarities in terms of the type of
information, mathematical approach, and features the build on or offer. A
subsequent contribution is a discussion of mechanisms to extend W-RBL
techniques to soft-connected rigid body localization (SCW-RBL) algorithms
Optimal Initialization Strategies for Range-Only Trajectory Estimation
Range-only (RO) pose estimation involves determining a robot's pose over time
by measuring the distance between multiple devices on the robot, known as tags,
and devices installed in the environment, known as anchors. The nonconvex
nature of the range measurement model results in a cost function with possible
local minima. In the absence of a good initialization, commonly used iterative
solvers can get stuck in these local minima resulting in poor trajectory
estimation accuracy. In this work, we propose convex relaxations to the
original nonconvex problem based on semidefinite programs (SDPs). Specifically,
we formulate computationally tractable SDP relaxations to obtain accurate
initial pose and trajectory estimates for RO trajectory estimation under static
and dynamic (i.e., constant-velocity motion) conditions. Through simulation and
real experiments, we demonstrate that our proposed initialization strategies
estimate the initial state accurately compared to iterative local solvers.
Additionally, the proposed relaxations recover global minima under moderate
range measurement noise levels
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
SCORE: A Second-Order Conic Initialization for Range-Aided SLAM
We present a novel initialization technique for the range-aided simultaneous
localization and mapping (RA-SLAM) problem. In RA-SLAM we consider measurements
of point-to-point distances in addition to measurements of rigid
transformations to landmark or pose variables. Standard formulations of RA-SLAM
approach the problem as non-convex optimization, which requires a good
initialization to obtain quality results. The initialization technique proposed
here relaxes the RA-SLAM problem to a convex problem which is then solved to
determine an initialization for the original, non-convex problem. The
relaxation is a second-order cone program (SOCP), which is derived from a
quadratically constrained quadratic program (QCQP) formulation of the RA-SLAM
problem. As a SOCP, the method is highly scalable. We name this relaxation
Second-order COnic RElaxation for RA-SLAM (SCORE). To our knowledge, this work
represents the first convex relaxation for RA-SLAM. We present real-world and
simulated experiments which show SCORE initialization permits the efficient
recovery of quality solutions for a variety of challenging single- and
multi-robot RA-SLAM problems with thousands of poses and range measurements.Comment: 9 pages, 8 figures, extended version of paper submitted to ICRA 202
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