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Register a new userAnalysis of minima for geodesic and chordal cost for a minimal 2D pose-graph SLAM problem
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Added by
Felix Kong
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we show that for a minimal pose-graph problem, even in the ideal case of perfect measurements and spherical covariance, using the so-called wrap function when comparing angles results in multiple suboptimal local minima. We numerically estimate regions of attraction to these local minima for some numerical examples, and give evidence to show that they are of nonzero measure. In contrast, under the same assumptions, we show that the textit{chordal distance} representation of angle error has a unique minimum up to periodicity. For chordal cost, we also search for initial conditions that fail to converge to the global minimum, and find that this occurs with far fewer points than with geodesic cost.
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We present a first step towards a multigrid method for solving the min-cost flow problem. Specifically, we present a strategy that takes advantage of existing black-box fast iterative linear solvers, i.e. algebraic multigrid methods. We show with standard benchmarks that, while less competitive than combinatorial techniques on small problems that run on a single core, our approach scales well with problem size, complexity, and number of processors, allowing for tackling large-scale problems on modern parallel architectures. Our approach is based on combining interior-point with multigrid methods for solving the nonlinear KKT equations via Newtons method. However, the Jacobian matrix arising in the Newton iteration is indefinite and its condition number cannot be expected to be bounded. In fact, the eigenvalues of the Jacobian can both vanish and blow up near the solution, leading to a significant slow-down of the convergence speed of iterative solvers - or to the loss of convergence at all. In order to allow for the application of multigrid methods, which have been originally designed for elliptic problems, we furthermore show that the occurring Jacobian can be interpreted as the stiffness matrix of a mixed formulation of the weighted graph Laplacian of the network, whose metric depends on the slack variables and the multipliers of the inequality constraints. Together with our regularization, this allows for the application of a black-box algebraic multigrid method on the Schur-complement of the system.
We present a simultaneous localization and mapping (SLAM) algorithm that is based on radio signals and the association of specular multipath components (MPCs) with geometric features. Especially in indoor scenarios, robust localization from radio signals is challenging due to diffuse multipath propagation, unknown MPC-feature association, and limited visibility of features. In our approach, specular reflections at flat surfaces are described in terms of virtual anchors (VAs) that are mirror images of the physical anchors (PAs). The positions of these VAs and possibly also of the PAs are unknown. We develop a Bayesian model of the SLAM problem and represent it by a factor graph, which enables the use of belief propagation (BP) for efficient marginalization of the joint posterior distribution. The resulting BP-based SLAM algorithm detects the VAs associated with the PAs and estimates jointly the time-varying position of the mobile agent and the positions of the VAs and possibly also of the PAs, thereby leveraging the MPCs in the radio signal for improved accuracy and robustness of agent localization. The algorithm has a low computational complexity and scales well in all relevant system parameters. Experimental results using both synthetic measurements and real ultra-wideband radio signals demonstrate the excellent performance of the algorithm in challenging indoor environments.
Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear systems. Chordal decomposition exploits the sparsity of semidefinite matrices in a semidefinite program (SDP), in order to formulate an equivalent SDP with smaller semidefinite constraints that can be solved more efficiently. Factor-width decompositions, instead, relax or strengthen SDPs with dense semidefinite matrices into more tractable problems, trading feasibility or optimality for lower computational complexity. This article reviews recent advances in large-scale semidefinite and polynomial optimization enabled by these two types of decomposition, highlighting connections and differences between them. We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine learning. Finally, we outline possible directions for future research that have the potential to facilitate the efficient optimization-based study of increasingly complex large-scale dynamical systems.
Nowadays in the field of semantic SLAM, how to correctly use semantic information for data association is still a problem worthy of study. The key to solving this problem is to correctly associate multiple object measurements of one object landmark, and refine the pose of object landmark. However, different objects locating closely are prone to be associated as one object landmark, and it is difficult to pick up a best pose from multiple object measurements associated with one object landmark. To tackle these problems, we propose a hierarchical object association strategy by means of multiple object tracking, through which closing objects will be correctly associated to different object landmarks, and an approach to refine the pose of object landmark from multiple object measurements. The proposed method is evaluated on a simulated sequence and several sequences in the Kitti dataset. Experimental results show a very impressive improvement with respect to the traditional SLAM and the state-of-the-art semantic SLAM method.
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